An Assessment of the Inherent Reliability of SANS 10162-2

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An Assessment of the Inherent Reliability of SANS 10162-2 for

Cold-Formed Steel Columns using the Direct Strength Method

Presented By:

Michael Alexander West-Russell

Thesis presented in partial fulfilment of the requirements for the degree of Master of Structural Engineering in the Faculty of Civil Engineering at Stellenbosch University

Supervisors:

Dr. C. Viljoen

Mr. E. Van der Klashorst

March 2018

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Declaration: By submitting this thesis electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly otherwise stated), that reproduction and publication thereof by Stellenbosch

University will not infringe any third-party rights and that I have not previously in its

entirety or in part submitted it for obtaining any qualification.

Signed: .................................................... Michael Alexander West-Russell

Date: .......................................................

Copyright © 2017

Stellenbosch University

All Rights Reserved

Stellenbosch University https://scholar.sun.ac.za

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Abstract With their thin and slender nature, Cold formed Steel (CFS) elements can be forged into a large variety of cross sections. It is recommended that the Direct Strength Method (DSM), a modern design method, replaces the effective width method for the design of

CFS members.

Research by Bauer (2016) revealed that, using the DSM, there is an insufficient level of

safety provided by SANS 10162-2: design guide for Cold-Formed Steel Structures to achieve the target reliability prescribed by SANS 10160-1: the basis of structural design.

This is partly due to a relatively high model uncertainty of the structural resistance. The

main objective of this study is to determine whether the level of safety provided by SANS 10160-1 is enough to ensure an overall acceptable level of reliability for CFS members.

A prequalified plain lipped C-section is considered for the investigation of the local and

global buckling failure modes. A prequalified lipped C-section with a web stiffener is

considered for the investigation of the distortional and global buckling failure modes. Various member lengths are considered to ensure that all buckling modes are induced.

The Finite Strip Method (FSM) is performed to identify the three buckling modes.

The members are subject to four load cases. Of which, each had different combinations of

permanent, imposed and wind loads determined by partial factors and combination factors presented in SANS 10160-1. For the purposes of this study, only the ultimate limit-

state was considered.

The considered limit-state is analysed in two parts; the semi- and full probabilistic formulations of the limit-state. The former considers the structural resistance and the

load effect according to codified design. The latter considers the structural resistance and

the load effect as functions of random variables. The reliability analysis is conducted on the full probabilistic formulation of the limit-state to assess the reliability achieved by

using the codified semi-probabilistic formulation of the limit-state.

Model factors are considered for the structural resistance and the load effect. The

structural resistance model factor is dependent on the buckling mode (Ganesan and Moen, 2010). A model factor for the permanent load is not considered and the model factor for


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wind load had been accounted for in the respective statistical moment parameters. A model factor for the imposed load is described by Holický (2009).

The statistical moment parameters of the permanent and imposed loads are described by

Holický (2009) and the statistical moment parameters of the wind load are described by

Botha (2016).

The global buckling mode yields the lowest reliability levels, ranging from β = 1.78

corresponding to an STR-P load combination to β = 2.87 corresponding to an STR load combination. The safety margin present in SANS 10160-1 partially compensates for the

low level of reliability when the total load comprises of high proportions of imposed and

wind load. The low levels of reliability, especially when there is significant permanent loading, are cause for concern. It is recommended that a different capacity reduction factor

be applied to each dominating buckling mode for CFS compression members.


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Acknowledgements The completion of this thesis would not have been possible if it were not for the encouragement and influence of several people. Their contribution and support have been more than I could have ever expected.

Above all, I would like to express great appreciation to my study supervisors, both of

whom provided funding during my postgraduate studies.

Since introducing me to the prospect of continuing my studies to a postgraduate degree,

Dr. Celeste Viljoen has kept me excited to learn and motivated to work hard. As a co-supervisor, her deep understanding of reliability theory has truly been an enormous

contributor to the basis of this work.

Before his departure from Stellenbosch, Mr. Etienne Van der Klashorst had consistently

been supportive of my efforts and patient with my learning as a co-supervisor. His profound knowledge of the behaviour of structural steel members has been imperative to

the success of this study.

Furthermore, I would like to thank my family. Throughout my entire academic career,

they have never ceased to offer their encouragement. Consistently willing to share in my stresses and successes, the support structure that they offer is never failing.

Lastly, I would like to thank the 2017 MEng Structural Design graduate students.

Everyone has been prepared to discuss ideas and share valuable opinions. Continuing our

studies to a postgraduate level, they have ensured that my student life will never be forgotten, both in and out of the office.


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Table of Contents Declaration: ............................................................................................................................. i

Abstract .................................................................................................................................. ii

Acknowledgements ............................................................................................................... iv

Table of Contents ................................................................................................................... v

List of Figures ..................................................................................................................... viii

List of Tables .......................................................................................................................... x

Nomenclature ...................................................................................................................... xiii

CHAPTER 1: Introduction .................................................................................................... 1

1.1 Background ............................................................................................................. 1

1.2 Problem Statement ................................................................................................ 2

1.3 Aim of study ............................................................................................................ 2

1.4 Limitations ............................................................................................................. 3

CHAPTER 2: Background into Cold-Formed Steel and Reliability Theory ....................... 4

2.1 Cold-Formed Steel .................................................................................................. 4

2.1.1 Description and Application ............................................................................ 4

2.1.2 Design Methods of Cold-Formed Steel Members ............................................ 6

2.2 Reliability Theory ................................................................................................. 18

2.2.1 The Ultimate Limit-State Equation .............................................................. 18

2.2.2 Reliability Simulation Methods ..................................................................... 26

2.3 Model Factor ......................................................................................................... 33

CHAPTER 3: Assessment of Cold-Formed Steel Columns Designed with the Direct

Strength Method .................................................................................................................. 37

3.1 Basis of Reliability Analysis ................................................................................ 37

3.1.1 Semi-Probabilistic Formulation of the Limit-State ...................................... 38


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3.1.2 Full Probabilistic Formulation of the Limit-State ....................................... 39

3.1.3 Process of Reliability Analysis ....................................................................... 40

3.2 Analysis of Cold-Formed Steel Compression Member ....................................... 43

3.2.1 Member Geometric Properties ....................................................................... 43

3.2.2 Member Material Properties ......................................................................... 46

3.2.3 Design and Mean Yield Loads ....................................................................... 47

3.2.4 Signature Curves and Buckling Modes ......................................................... 48

3.2.5 Direct Strength Method Design of Members ................................................ 51

3.2.6 Reliability Analyses ........................................................................................ 54

3.3 Semi-Probabilistic Formulation of the Limit-State ............................................ 56

3.3.1 Obtaining Characteristic Values of the Loads .............................................. 56

3.3.2 Load Combination Choice .............................................................................. 59

3.4 Full Probabilistic Formulation of the Limit-State ............................................. 62

3.4.1 Variables of the Load Effect ........................................................................... 62

3.4.2 Variables of the Structural Resistance ......................................................... 64

3.4.3 Reliability Analysis ........................................................................................ 65

3.4.4 Monte-Carlo Simulation ................................................................................. 66

3.4.5 Chapter Summary .......................................................................................... 66

CHAPTER 4: Results & Discussion .................................................................................... 68

4.1 Results of Reliability Analysis ............................................................................. 68

4.1.1 Global Buckling .............................................................................................. 68

4.1.2 Local Buckling and Local-Global Buckling Interaction ............................... 81

4.1.3 Distortional Buckling ..................................................................................... 89

4.2 Discussion of Reliability Results ......................................................................... 94

4.2.1 Overview ......................................................................................................... 94

4.2.2 Discussion of Reliability Levels of Buckling Modes ..................................... 94

4.2.3 Detailed Discussion ........................................................................................ 95


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CHAPTER 5: Conclusion & Recommendations ................................................................. 97

5.1 Conclusion ............................................................................................................ 97

5.2 Recommendations ................................................................................................ 99

References .......................................................................................................................... 101


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List of Figures Figure 2.1: Various sections of cold-formed steel (Yu & LaBoube, 2010) ........................... 5

Figure 2.2: Edge loaded simply supported plate (American Iron and Steel Institute et al.,

2016) ....................................................................................................................................... 7

Figure 2.3: Change in stress distribution with increasing edge load (Yu and Schafer, 2005)

................................................................................................................................................. 7

Figure 2.4: Representation of the local buckling mode (Bauer, 2016) ................................ 9

Figure 2.5: Representation of the distortional buckling mode (Bauer, 2016) .................... 9

Figure 2.6: Representation of the global buckling mode in the form of (a) flexural buckling,

(b) torsional buckling and (c) flexural-torsional buckling (Bauer, 2016) .......................... 10

Figure 2.7: Typical signature curve showing the three types of buckling modes (Bauer,

2016) ..................................................................................................................................... 12

Figure 2.8: Probability density functions of load effect and structural resistance (Bauer, 2016) ..................................................................................................................................... 21

Figure 2.9: Probability density function of the limit-state G (Bauer, 2016) ..................... 22

Figure 2.10: Two-dimensional space of state variables (Nowak and Collins, 2000) ........ 24

Figure 2.11: 3-Dimentional plot of the joint density function (Nowak and Collins, 2000)

............................................................................................................................................... 24

Figure 2.12: Reliability margin and sensitivity factors in a standardised space for a non-

linear limit-state equation (Haldar and Mahadevan, 2000) .............................................. 25

Figure 2.13: First order reliability method (Holický, 2009) .............................................. 27

Figure 2.14: Graphical representation of the FORM analysis (Lopez and Beck, 2012) ... 29

Figure 2.15: Test-to-predicted ratios of CFS columns as a function of global slenderness

for the DSM (Ganesan and Moen, 2010) ............................................................................. 34

Figure 2.16: General concept of the model factor (Holický et al, 2015) ............................ 36

Figure 3.1: Sensitivity factors of structural resistance random variables (Bauer, 2016) 38

Figure 3.2: Flow diagram depicting the general procedure of the reliability analysis .... 42


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Figure 3.3: Plain lipped C-section with centreline dimensions ......................................... 44

Figure 3.4: Stiffened lipped C-section with centreline dimensions ................................... 44

Figure 3.5: Signature curves of plain lipped C-section ...................................................... 49

Figure 3.6: Signature curves of stiffened lipped C-section ................................................ 49

Figure 3.7: Direct strength method buckling loads for the plain lipped C-section .......... 52

Figure 3.8: Direct strength method buckling loads for the stiffened lipped C-section .... 53


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List of Tables Table 2.1: Pre-qualified geometric limits for compression members (SANS 10162-2, 2011)............................................................................................................................................... 14

Table 2.2: Categorisation of the model factor (Holicky, et al., 2015) ................................ 34

Table 3.1: Geometrical properties of compression elements ............................................. 45

Table 3.2: Sectional properties obtained from CUFSM ..................................................... 46

Table 3.3: Material properties of Grade G550 high strength steel ................................... 47

Table 3.4: Yield stresses and yield capacities for the plain and stiffened lipped C-sections............................................................................................................................................... 48

Table 3.5: Load factors of Nd and Nµ signature curves for plain lipped-C section ........... 50

Table 3.6: Load factors of Nd and Nµ signature curves for stiffened lipped-C section ..... 50

Table 3.7: Test summary of reliability analysis ................................................................. 55

Table 3.8: Summary of considered load combinations ....................................................... 57

Table 3.9: Values of partial and combination factors ........................................................ 57

Table 3.10: Dominating load combinations for different values of χQ and χW .................. 61

Table 3.11: Statistical moment parameters of the considered loads (Holický, 2009) ...... 62

Table 3.12: Updated Eurocode full probabilistic wind load model of the FORM analysis (Botha, 2016) ........................................................................................................................ 63

Table 3.13: Statistical moment parameters of the structural resistance model factors for different buckling modes (Ganesan & Moen, 2010) ........................................................... 65

Table 3.14: MC analyses of dominating load combinations ............................................... 66

Table 4.1: Reliability levels for test one .............................................................................. 69

Table 4.2: α𝜕R for test one ..................................................................................................... 70

Table 4.3: α𝜕Q for test one .................................................................................................... 70

Table 4.4: αG for test one ...................................................................................................... 71


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Table 4.5: αQ for test one ...................................................................................................... 71

Table 4.6: αW for test one ..................................................................................................... 72

Table 4.7: Reliability levels for test two ............................................................................. 73

Table 4.8: α𝜕R for test two .................................................................................................... 74

Table 4.9: α𝜕Q for test two .................................................................................................... 74

Table 4.10: αG for test two ................................................................................................... 75

Table 4.11: αQ for test two ................................................................................................... 75

Table 4.12: αW for test two ................................................................................................... 76

Table 4.13: Reliability levels for test three ......................................................................... 77

Table 4.14: α𝜕R for test three ................................................................................................ 77

Table 4.15: α𝜕Q for test three ............................................................................................... 78

Table 4.16: αG for test three ................................................................................................. 78

Table 4.17: αQ for test three ................................................................................................. 79

Table 4.18: αW for test three ................................................................................................ 79

Table 4.19: Reliability levels achieved through FORM and MCS for each load combination

............................................................................................................................................... 81

Table 4.20: Reliability levels for test four ........................................................................... 82

Table 4.21: α𝜕R for test four .................................................................................................. 82

Table 4.22: α𝜕Q for test four ................................................................................................. 83

Table 4.23: αG for test four ................................................................................................... 83

Table 4.24: αQ for test four ................................................................................................... 84

Table 4.25: αW for test four .................................................................................................. 84

Table 4.26: Reliability levels for test five ........................................................................... 85

Table 4.27: α𝜕R for test five .................................................................................................. 86

Table 4.28: α𝜕Q for test five .................................................................................................. 86

Table 4.29: αG for test five ................................................................................................... 87

Table 4.30: αQ for test five ................................................................................................... 87


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Table 4.31: αW for test five ................................................................................................... 88

Table 4.32: Reliability levels for test six ............................................................................. 90

Table 4.33: α𝜕R for test six .................................................................................................... 90

Table 4.34: α𝜕Q for test six ................................................................................................... 91

Table 4.35: αG for test six ..................................................................................................... 91

Table 4.36: αQ for test six ..................................................................................................... 92

Table 4.37: αW for test six .................................................................................................... 92


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Nomenclature Acronyms

AISI American Iron and Steel Institute

CFS Cold-Formed Steel

cFSM Constrained Finite Strip Method

DSM Direct Strength Method

FORM First Order Reliability Method

FSM Finite Strip Method

MCS Monte Carlo Simulation

SANS South African National Standard

VaP Variable Processor

Symbols

[A] Load factor matrix

{α} Sensitivity factor vector

{D} Vector of partial derivatives

{Nk} Characteristic load vector

{Nd} Design load vector

{U} Vector of standardised random variables

{u*} Vector of standardised random variable realisations

{X} Vector of random variables

{x*} Vector of random variable realisations

α Sensitivity factor


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α𝜕 Sensitivity factor of the model uncertainty

α𝜕Q Sensitivity factor of the imposed load model factor

α𝜕R Sensitivity factor of the structural resistance model factor

αE Sensitivity factor of the load effect

αG Skewness of the limit state probability density distribution

αR Sensitivity factor of the structural resistance

β Reliability index

βf Stress coefficient

βt Target reliability index

𝛾G Partial factor of the permanent load

𝛾Q Partial factor of the imposed load

𝛾W Partial factor of the wind load

𝜕 Model factor

𝜕E Model factor of the load effect

𝜕R Model factor of the structural resistance

𝜕R(d) Model factor for the distortional buckling mode

𝜕R(g) Model factor for the global buckling mode

𝜕R(lg) Model factor for the local-global interaction buckling mode

θ Lip angle

λc Non-dimensional slenderness for global buckling of a column

λd Non-dimensional slenderness for distortional buckling of a column

λl Non-dimensional slenderness for local buckling of a column

σ Standard deviation

σE Standard deviation of the load effect

σe Standarddeviationoftheequivalentnormaldistribution


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σR Standard deviation of the structural resistance

σX Standard deviation of a random variable

µ Meanvalue

µE Meanvalueoftheloadeffect

µe Meanvalueoftheequivalentnormaldistribution

µR Meanvalueofthestructuralresistance

µX Meanvalueofarandomvariable

ρRE Correlation coefficient of structural resistance and load effect

ΦG Probability distribution function of the limit-state

ΦU Probability distribution function of a standardised random variable

φ Probability density function

φc Capacity reduction factor for compression elements

χQ Imposed load ratio

χW Wind load ratio

A Cross-sectional area

t Thickness

E Modulus of elasticity (Young’s Modulus)

E Load effect

Ed Codified design value of the load effect.

foc Elastic flexural, torsional, and flexural-torsional buckling stress

fod Elastic distortional buckling stress

fol Elastic local buckling stress

fox Elastic buckling stress in a compressive element for flexural buckling about

the x-axis

foxz Elastic buckling stress in a compressive element for flexural-torsional

buckling about the x-axis


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foy Elastic buckling stress in a compressive element for flexural buckling about the y-axis

foz Elastic buckling stress in a compressive element for torsional buckling

fy Yield stress

G Limit-state

G Shear modulus

Gd Design value of the permanent load

Gk Characteristic value of the permanent load

h Web height

Iw Warping constant for a cross-section

J Torsion constant for a cross-section

L Member length

l Lip height

le Effective member length

lex Effective member length for buckling about the x-axis

ley Effective member length for buckling about the y-axis

lez Effective member length for twisting

ld Distortional buckling half-wavelength

ll Local buckling half-wavelength

ll-g Local-Global interaction buckling half-wavelength

Nµ Mean compressive critical buckling load

Ncl Nominal compressive local buckling capacity

Ncd Nominal compressive distortional buckling capacity

Nce Nominal compressive global buckling capacity

Ncr,d Distortional buckling based compressive capacity of a column


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Ncr,e Least of the elastic flexural, torsional, and flexural-torsional buckling compressive load

Ncr,l Local buckling based compressive capacity of a column

Nd Design compressive critical buckling load

Ny Compressive yield load

Ny,µ Yield capacity based on the design values of material properties

Ny,d Yield capacity based on the mean values of material properties

pf Probability of failure

R Structural resistance

Rd Codified design value of the structural resistance.

r Lip radius or radius of gyration

ro1 Polar radius of gyration of the cross section about the shear centre

rx, ry Radius of gyration about the x- and y-axes

U Standardised random variable

u Realisation of a standardised random variable

u* Standardised design point

V Coefficient of Variation

ν Poisson’s Ratio

w Flange width

X Random variable

XE Random variable associated with the load effect

XG Random variable of the permanent load

XQ Random variable of the imposed load

XR Random variable associated with the load effect

XW random variable of the wind load


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x Realisation of a random variable

x* Design point

xcg, ycg Coordinates of the centre of gravity of a cross-section

xo, yo Coordinates of the shear centre of a cross-section


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1 CHAPTER 1: Introduction

1.1 Background When a design method is introduced to codified design standards, the reliability level of the method needs to be investigated. Various probability analyses are required to

determine the level of inherent reliability of a newly developed design method.

Considering the variables involved in structural design, the influence that the variables have on the reliability also needs to be investigated. The target level of reliability depends

on direct and indirect costs of failure such as loss of human life, economic, social, and

environmental consequences (Holický, 2009).

As a construction material, thin-walled steel has been used in a range of structures. Failure of the thin-walled steel members in these structures may result in the loss of

human life, pollution, or damage to property. The use of thin-walled elements as a

construction material is increasing, as design methods are becoming more simplified and the use of steel is further optimised. With the growing application of thin-walled steel

members, it is becoming more important to ensure the safety of the material in structural

design.

Proposed by Schafer (2006a), a design method called the Direct Strength Method (from here on referred to as DSM) has been introduced to thin-walled steel design over the last

two decades. The method was officially adopted as an appendix to the American Iron and Steel Institute (from here on referred to as AISI) design code for Cold-Formed Steel (from

here on referred to as CFS) structural members in 2004 and as an additional chapter to

SANS 10162-2: Cold-formed steel structures (from here on referred to as SANS 10162-2). The method incorporates the unique structural behaviour of the thin-walled steel

elements with existing codified design procedures.


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1.2 Problem Statement SANS 10162-2 (2011) uses partial factors to provide a safety margin. The code is an

adaption of the Australian/New Zealand standard (AS/NZS 4600) for CFS members in structural design. The target reliability level presented in the South African National

Standard 10160-1: Basis of structural design (from here on referred to as SANS 10160-1) for reliability class 2 structures is a factor of βt = 3. Reliability verification procedures are

generally specified as reliability class 2 structures as a reference classification.

The work conducted by Bauer (2016) examined the inherent reliability of the DSM in a

South African design context, particularly for the structural resistance of CFS elements.

Amongst the findings of the study, two points are to be noted. Firstly, it was found that the level of model uncertainty for the structural resistance had a dominating effect on the

level of reliability. Secondly, the safety margin of the structural resistance provided by

SANS 10162-2 (2011) proved insufficient to achieve the target reliability level. The study, however, did not consider the effect the load effect had on the reliability level.

1.3 Aim of study The primary goal of this study is to determine whether there is a sufficient safety margin

supplied by SANS 10160-1 (2011) to achieve an overall acceptable level of reliability for CFS compression members. The compression members are to be designed in accordance

to the DSM. The reliability levels of each dominating buckling mode presented by the

DSM are to be determined.

The reliability analysis needs to be conducted on the full probabilistic formulation of the limit-state to assess the inherent reliability of the codified semi-probabilistic formulation

of the limit-state. The structural resistance and the load effect of the full probabilistic

formulation of the limit state are to be expressed as functions of random variables.

The reliability analysis needs to consider various dominating load combinations. The total load needs to be comprised of the permanent, imposed and the wind load. In the analysis,

load combinations of SANS 10160-1 (2011) are to be considered over a full range of ratios

of characteristic imposed loads and wind loads to the characteristic total load.

Once the above is achieved, the reliability levels produced from the full probabilistic

analysis of the limit state must be assessed and scrutinised for all buckling modes by

varying member lengths. Additionally, the sensitivity factors for each of the associated


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random variables must be evaluated to assess their respective influences on the level of reliability.

1.4 Limitations Innate to the use of the DSM are certain limitations. Only prequalified members may be

used to adhere to the conventional capacity reduction factor presented in SANS 10162-2 (2011). This is due to the limits of available research and development (Schafer, 2006a).

This includes a limitation on the ratio of the Elastic modulus to the yield strength of the

steel, as well as cross-sectional geometric limitations.

The DSM does not make provision for cross sections that vary along the length of the member. This includes members with holes and elements that are tapered. Additionally,

the applied load must be consistent throughout the length of the member.

The DSM makes provision for compression and bending CFS elements. For the purposes

of this study, only compression members were considered.

Four load combinations were considered. Of each, only the permanent, imposed and wind loads were included.


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2 CHAPTER 2: Background into Cold-Formed

Steel and Reliability Theory

The following chapter covers the fundamental information required to understand the

behaviour of cold-formed steel sections and how design methods incorporate this into design equations. Relevant reliability theory is discussed for the purposes of this

investigation and how it is applied to cold-formed steel members.

2.1 Cold-Formed Steel 2.1.1 Description and Application

CFS is a light-weight and slender building material. According to Yu and LaBoube (2010),

the manufacturing processes of cold-formed sections begin from rolls, plates, or strips of steel. The steel makes its way through roll-forming machines at room temperature. The

manufacturing process of CFS members enables it to be formed into a variety of cross

sections. The typical cross-sections that are widely used in the South African construction industry are C-, Z, and hat-sections. Each of these sections can be stiffened or lipped.

Examples of CFS cross-section are shown in Figure 2.1 from Yu and LaBoube (2010). The

plate thickness of the members typically ranges between 0.378mm to 6.35mm, depending on the required application. With a relatively high width-to-thickness ratio, CFS members

are considered as thin-walled slender members.


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Figure 2.1: Various sections of cold-formed steel (Yu & LaBoube, 2010)

Due to their slender nature, the behaviour of thin-walled members under loads are different from conventional hot-rolled steel members. As the thickness of a member

decreases, the elements of the member begin to buckle in a similar manner to plate

buckling. Therefore, one cannot use the same design principles of thin-walled members using the limit-states design method presented in South African National Standard

10162-1: Limit-states design of hot-rolled steelwork (from here on referred to as SANS

10162-1).

Although cold-formed steel is not as familiar as hot-rolled steel, cold-formed steel is growing in application, importance, and innovation in the construction industry. When

CFS was first introduced as a building material in the 1930’s, the acceptance of it in the

construction industry was limited, since there were no adequate design standards or methods (Yu and LaBoube, 2010). Their use ranged from truss and wall panel systems to

structural bracing and roof purlins. However, these types of elements were rarely used as

primary load-bearing elements such as columns and beams (Allen, 2006).

The application of this type of steel has since evolved and buildings up to six storeys consisting of primarily cold-formed steel framing elements have been constructed (Yu and

LaBoube, 2010). This type of steel is not only more cost-effective and lighter than hot rolled steel, but requires less labour to erect structures. Thus, it is an attractive

construction material.

However, the conventional CFS design method may dissuade designers from optimising

the cross-section of elements, as the design procedure is complex and iterative. This limits the potentially broad application of the steel. Thus, a new design method called the Direct

Strength Method has been developed to address these design issues (Schafer, 2006b).


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In the design context of South Africa, the reliability level achieved when using the DSM for CFS members needs to be assessed. Freitas et al. (2013) conducted a study that

reviewed the calibration procedure of the capacity reduction factor for CFS compression

members using the DSM in order to achieve a target reliability. The target reliability was achieved by using a capacity reduction factor of 0.88; higher than that of 0.85 suggested

by SANS 10162-2 (2011). However, the target reliability used in the study was of βt = 2.5

and not of βt = 3. Therefore, the current capacity reduction factors suggested by SANS

10162-2 (2011) may not be sufficient to achieve a target reliability of βt = 3. Additionally,

Freitas et al. (2013) only considered two load combinations; both of which exclude the effect of the wind load.

2.1.2 Design Methods of Cold-Formed Steel Members

Presented in SANS 10162-2 (2011) are two options for CFS compression member design.

The main design specification for CFS compression members is the Effective Width Method (from here on referred to as EWM). Accompanying this method as a separate

chapter of SANS 10162-2 (2011) is the DSM. Both methods are subsequently discussed.

This is to highlight the differences in design approaches of these methods. With both methods having strengths and limitations, motivation is given for the Direct Strength

Method to be the preferred method of CFS member design.

2.1.2.1 Effective Width Method The conventional method used to design CFS compression members in SANS 10162-2

(2011) is the EWM, developed by the AISI (Ziemain, 2010). Simply put, the design equations of the EWM incorporates plate buckling theory. As a compressive stress

increases on the edge of a simply supported plate, the plate begins to buckle before the

yield stress is reached. Additionally, it its said that an edge of an element is restrained if the element is joined to another element. Due to the restrained edges of CFS members,

there is still some post-buckling strength to account for.

Consider a simply supported plate that is edge loaded with a compressive stress shown in

Figure 2.2 from American Iron and Steel Institute et al. (2016). Initially, the stress distribution across the edge of the plate is uniform through the width of the plate before

buckling occurs. As the stress increases, the plate begins to buckle. At this point, the

stress begins to redistribute towards the restrained edges of the plate, causing a non-uniform stress distribution. If the edges of the plate were not restrained, then the capacity


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of the plate would be determined by the idealised elastic critical buckling stress of the plate (American Iron and Steel Institute et al., 2016).

Figure 2.2: Edge loaded simply supported plate (American Iron and Steel Institute et al., 2016)

As the applied stress along the loaded edge increases, the stress at the corners of the plate

also continue to increase with further stress distribution. Finally, failure occurs when the stress at the edges of the plate reach the material yield stress. The development of the

stress distribution along the loaded edge of the plate is shown in Figure 2.3 by Yu and

Schafer (2005). The phenomenon of the non-linear stress distribution towards the edges is known as the post-buckling strength (Yu and LaBoube, 2010).

Figure 2.3: Change in stress distribution with increasing edge load (Yu and Schafer, 2005)

Observing the stages of the stress distribution in Figure 2.3, part (a) is where the applied uniform stress is less than the critical buckling load. Part (b) is where the redistribution

of the stress towards the edges begins. Part (c) is where the most stress distribution has

occurred, as the outer edges begin to yield.

This method suggests that the resistance of the entire plate is represented by a part of the width of the plate only; the effective width of the section. The effective width comprises

w

w d

a

c

b


8

of edge parts, where fully plastic stresses can occur. As CFS members can consist of many such elements, the overall resistance is calculated as the sum of all resistances of the

individual compressed plate elements within the section of the member.

Although the Effective Width Method has been successfully used for CFS design since the

1940’s, the method does have drawbacks. With complex CFS sections, this method becomes a tedious procedure with cumbersome iterations. Furthermore, the EWM

idealises elastic buckling as individual elements, which ignores the interaction between

plate elements (Schafer, 2002).

2.1.2.2 Direct Strength Method Schafer (2006a) states that the Direct Strength Method was formally adopted as an Appendix in the North American Specification for the Design of Cold-Formed Steel

Structural Members in 2004. This newly adopted method attempts to meet current design

needs of simple and complex CFS sections through simplified means. SANS 10162-2 (2011) includes the method as an alternative procedure to design thin-walled columns and

beams.

Incorporated within the DSM is the Finite Strip Method (from here on referred to as

FSM). This method is closely linked to plate buckling theory. The method identifies the buckling modes, as well as the capacity of each buckling mode of any thin-walled element

to be assessed. One should not confuse the DSM with the FSM, as the FSM is presented as a procedural calculation step within the DSM.

2.1.2.2.1 Buckling Modes Through the development of the DSM, three main stability modes were defined, namely local, distortional, and global buckling modes. The definitions of each are subsequently

given by Schafer (2006a).

Local Buckling – This buckling mode induces rotational distortion of the cross section at the corners of the member. This is to say that the angles between the elements do not

change whilst the elements within the section buckle. There is no translational distortion

of the section for this buckling mode. Under a compressive load, the buckling half-wavelength of this buckling mode is no greater than the largest dimension of the member.

An illustration of this buckling mode for a lipped channel section is shown in Figure 2.4

by Bauer (2016).


9

Figure 2.4: Representation of the local buckling mode (Bauer, 2016)

Distortional Buckling – The distortional buckling mode induces rotational and translational distortion at one or more of the corners of the member. This means that the

angle between the elements change as the elements rotate about their intersections.

Although this buckling mode is load and geometry dependant, the buckling half-wavelength is larger than that of local buckling but smaller than that of global buckling.

This buckling mode is illustrated from Bauer (2016) in Figure 2.5 of a lipped channel section.

Figure 2.5: Representation of the distortional buckling mode (Bauer, 2016)

Global Buckling – Associated with flexural, torsional, or flexural-torsional buckling, the

global buckling mode for columns induces translation out of plane and/or rotation about

the shear centre of the cross section for axially loaded members. Unlike the local and distortional buckling modes, there is no deformation of the cross-sectional shape. The

buckling half-wavelength of this buckling mode depends on the end fixities of the member.


10

When pin-ended, the half-wavelength is at a maximum of the physical length of the member (Schafer, 2006b). From Bauer (2016), Figure 2.6 shows the global buckling mode

of a lipped channel section, demonstrating translation and rotation of the section.

Figure 2.6: Representation of the global buckling mode in the form of (a) flexural buckling, (b) torsional buckling and (c) flexural-torsional buckling (Bauer, 2016)

Buckling Mode Interactions – At certain member lengths, an interaction between

buckling modes can occur (Ungermann et al., 2014). The interaction of the buckling modes depends on the section geometry and profile. If an interaction between buckling modes

occur, the capacity of the member can be reduced. For the purposes of this study, only the

local-global buckling interaction was considered. This is because the test database that this research is based on only considered the local-global buckling interaction (Ganesan

& Moen, 2010).

2.1.2.2.2 Finite Strip Method As part of the DSM, the FSM must be used to determine unknown variables in the DSM

design procedure. Published in 1974 by Plank and Wittrick, the procedure of the Finite Strip Method allows for elastic behavioural prediction, requiring little effort (Ziemain,

2010). Without getting too detailed about the methodology of the Finite Strip Method, it

uses equations to approximate displacements along discrete strips along the length of the member.

To implement this method, the CUFSM v4.03 computer program was used. The program

is based in a MATLAB programming environment and has been made freely available

(a) (b) (c)


11

online by Prof. Benjamin Schafer at Johns Hopkins University in Baltimore, USA (Li and Schafer, 2010). The program requires certain input information about a specific section

that is to be analysed. The required input information of CUFSM includes geometrical

and material properties. The material properties required for input include the modulus

of elasticity (E), the Poison’s ratio (ν), the Yield stress (fy) and the shear modulus (G).

The section geometry is described using nodes and elements. This allows the user to input any shape of section described using as many elements and nodes as desired, with an

option to alter the thicknesses of the elements individually. To simplify the input procedure, templates for lipped C- and lipped Z-sections are supplied. The input values

for these templates are the centreline member height, flange widths, lip heights, lip radii

and sectional thickness. The program then determines the sectional properties of the section to be analysed.

The end-fixities and the length of the member must also be chosen. With the length of the

member, the half-wavelengths to be analysed should also be chosen. These half-

wavelengths are typically chosen to follow a logarithmic scale.

As explained in the Direct Strength Method Design Guide (Schafer, 2006a), the program produces two results through the Finite Strip Method; the buckling half-wavelength and

corresponding load factor, as well as the buckling mode shape of the cross section. The

buckling half-wavelength is an indication of how a given cross-sectional shape varies along the length of the member in the buckled state. The load factor is the ratio of the

critical design buckling load to the design yield load of the member in bending or in

compression and is shown in Equation 2.1.

𝐿𝑜𝑎𝑑𝐹𝑎𝑐𝑡𝑜𝑟 = 𝑁]^𝑁_

(2.1)

The results are displayed graphically in what is known as a signature curve. The shape of the signature curves differ for each CFS section of alternate geometric and material

properties analysed by the CUFSM program. The full member deformation signature

curve is comprised of three pure buckling mode signature curves of local, distortional, and global buckling.

These individual buckling mode signature curves are generated through the process

called the constrained finite strip method (cFSM) which CUFSM implements. An

important aspect of the cFSM process is that it identifies the contribution of each buckling


12

mode at each buckling half-wavelength. From Bauer (2016), a typical signature curve of a lipped C section under axial load is shown in Figure 2.7.

The curve shows that the local buckling region occurs at low buckling half-wavelengths.

The global buckling region occurs at high buckling half-wavelengths. The distortional

buckling zone occurs at buckling half-wavelengths between that of the local and global buckling zones.

The minima on the signature curve indicate the lowest load level at which each of the

buckling modes occur. Typically, there is a minimum point at the local and distortional

buckling zones of the curve. Here, the critical buckling load factors for the respective buckling modes may be found. Obtaining the load factors at the curve minima, the critical

buckling loads of Ncr,l and Ncr,d for local and distortional buckling respectively may then be calculated from Equation 2.1. This is on the condition that the compression yield load

Ny is known.

Figure 2.7: Typical signature curve showing the three types of buckling modes (Bauer, 2016)

𝐿𝑜𝑎𝑑𝐹𝑎𝑐𝑡𝑜𝑟𝑁

]^ 𝑁 _

𝐻𝑎𝑙𝑓𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ(𝑚𝑚)

𝑁]^,k𝑁_

𝑁]^,l𝑁_

𝑁]^,m𝑁_


13

2.1.2.2.3 Design Procedure of Compression Members Column design according to the DSM is formulated in Section 7 of SANS 10162-2 (2011). It states that the nominal capacity of a compression member (Nd) is taken as the minimum

of the capacity for local buckling (Ncl), distortional buckling (Ncd) and global buckling (Nce).

The codified design procedure includes a pre-qualification check. The member must fall

within the geometric limitations of the DSM to be designed using the corresponding

capacity reduction factor for compression members φc of 0.85 in the design calculation

procedure. A reduced capacity reduction factor for compression members φc of 0.80 is to

be used if the geometric limitations are not met. The pre-qualification requirements of

compression members for the DSM are listed in Table 2.1.


14

Table 2.1: Pre-qualified geometric limits for compression members (SANS 10162-2, 2011)

Section Geometric Limitation

Lipped Channel d/t < 472 b1/t < 159 4 < d1/t < 33 0.7 < d/b1 < 5.0 0.05 < d1/b1 < 0.41 θ = 90° E/fy > 340

Lipped Channel with Web Stiffener(s) d/t < 489 b1/t < 160 6 < d1/t < 33 1.3 < d/b1 < 2.7 0.05 < d1/b1 < 0.41 One or two intermediate stiffeners E/fy > 340

Z-Section d/t < 137 b1/t < 56 0 < d1/t < 36 1.5 < d/b1 < 2.7 0 < d1/b1 < 0.73 θ = 50° E/fy > 590

Rack Upright d/t < 51 b1/t < 22 5 < d1/t < 8 2.1 < d/b1 < 2.9 1.6 < b2/d1 < 2.0 (b2 = small outstand parallel to b1) d2/d = 0.3 (d2 = second lip parallel to d1) E/fy > 340

Hat d/t < 50 b1/t < 20 4 < d1/t < 6 1.0 < d/b1 < 1.2 d1/b1 = 0.13 E/fy > 428

r/t < 10, where r is the centre-line radius


15

Global Buckling Capacity

As per SANS 10162-2 (2011), the nominal global buckling capacity Nce of a member in compression is calculated as follows:

𝑁]m = 0.658stu∙ 𝑁_ 𝑓𝑜𝑟𝜆] ≤ 1.5 (2.2)

𝑁]m =0.877𝜆]

{ ∙ 𝑁_ 𝑓𝑜𝑟𝜆] > 1.5 (2.3)

where

𝜆] =𝑁_𝑁]^,m

(2.4)

𝑁_ = 𝐴 ∙ 𝑓_ (2.5)

𝑁]^,m = 𝐴 ∙ 𝑓~] (2.6)

λc is the non-dimensional slenderness for global buckling of the column. Ncr,e is the elastic

buckling load obtained from the lesser of the flexural, torsional, and flexural-torsional elastic buckling loads. A is the area of the full cross section. In the case for compression

members, foc is the least of the elastic flexural, torsional, and flexural-torsional buckling

stress. The procedure to determine foc is based on the calculation to determine the characteristic load of a concentrically loaded compression member.

For sections such as closed cross-sections that are not subject to torsional or flexural-

torsional buckling, foc is determined through Equation 2.7 from SANS 10162-2 (2011).

𝑓~] =𝜋{ ∙ 𝐸𝑙m 𝑟

{ (2.7)

E is the Young’s Modulus of Elasticity. le is effective length of the member. r is the radius

of gyration of the full, unreduced cross-section

For doubly- or singly-symmetric sections that are subject to torsional or flexural-torsional buckling, SANS 10162-2 (2011) states that foc is taken as the smaller of foc calculated using

Equation 2.7 with r as the radius of gyration about the weak axis and foxz calculated in

Equation 2.8.

𝑓~�� =1

2 ∙ 𝛽�∙ 𝑓~� + 𝑓~� − 𝑓~� + 𝑓~� { − 4 ∙ 𝛽 ∙ 𝑓~� ∙ 𝑓~� (2.8)


16

where

𝛽� = 1 −𝑥~𝑟~�

{(2.9)

𝑟~� = 𝑟�{ + 𝑟_{ + 𝑥~{ + 𝑦~{ (2.10)

βf is the stress coefficient, ro1 is the polar radius of gyration of the cross-section about the shear centre. rx and ry are the radii of gyration of the cross section about the x- and y-axes

respectively. xo and yo are coordinates of the shear centre of the cross section.

𝑓~� =𝜋{ ∙ 𝐸𝑙m� 𝑟�

{ (2.11)

𝑓~� =𝐺 ∙ 𝐽𝐴 ∙ 𝑟~�{

∙ 1 +𝜋{ ∙ 𝐸 ∙ 𝐼�𝐺 ∙ 𝐽 ∙ 𝑙m�

{ (2.12)

lex and lez are the effective lengths for buckling about the x-axis and for twisting

respectively. G is the shear modulus of elasticity. J is the torsion constant for the cross-section. Iw is the warping constant for the cross section.

For point-symmetric sections that are subject to flexural- or torsional buckling, SANS

10162-2 (2011) states that foc is taken as the lesser of foc as calculated in Equation 2.7 and foz as calculated in Equation 2.12.

Local Buckling Capacity

The nominal local buckling capacity Ncl of a member in compression is calculated as follows, from SANS 10162-2 (2011):

𝑁]l = 𝑁]m 𝑓𝑜𝑟𝜆l ≤ 0.776 (2.13)

𝑁]l = 1 − 0.15 ∙𝑁]^,l𝑁]m

�.�∙𝑁]^,l𝑁]m

�.�∙ 𝑁]m 𝑓𝑜𝑟𝜆l > 0.776 (2.14)

where

𝜆l =𝑁]m𝑁]^,l

(2.15)

λl is the non-dimensional slenderness for local buckling of the column and Ncr,l is the elastic local buckling load. The procedure to determine Ncr,l is explained in Section

2.1.2.2.2.


17

Distortional Buckling Capacity

SANS 10162-2 (2011) states that the nominal distortional buckling capacity Ncd of a member in compression is calculated as follows:

𝑁]k = 𝑁_ 𝑓𝑜𝑟𝜆k ≤ 0.561 (2.16)

𝑁]k = 1 − 0.25 ∙𝑁]^,k𝑁_

�.�

∙𝑁]^,k𝑁_

�.�

∙ 𝑁_ 𝑓𝑜𝑟𝜆k > 0.561 (2.17)

where

𝜆k =𝑁_𝑁]^,k

(2.18)

λd is the non-dimensional slenderness for distortional buckling of the column and Ncr,d is the elastic distortional buckling load. The procedure to determine Ncr,d is explained in

Section 2.1.2.2.2.


18

2.2 Reliability Theory The fundamental condition that all design methods are required to adhere to are certain limit-states. Holický (2009) states that limit-states are distinct conditions that separate

satisfactory and unsatisfactory states of a structure under certain conditions. Therefore,

if exceeded, the structure no longer satisfies specific design requirements.

Each limit-state is associated with specific performance criteria. It can become complex to quantify the performance criteria of the limit states. To simplify the design procedure,

two fundamental limit states for the design of structures are recognised: the ultimate and

serviceability limit-states.

The ultimate limit-state is a representation of the overall stability and safety of the structure. If the ultimate limit-state is exceeded, there is a loss of static equilibrium or

stability of the structure. These potential failure mechanisms need to be considered when

specifying the reliability parameters.

The serviceability limit-state corresponds to the result of normal use conditions such as a change in structural appearance and human comfort. It is important to take the time-

dependency of loads when considering the serviceability limit-state. If exceeded, cracks,

large deflections and noticeable vibrations may occur in a structure. Although it is important to consider both limit-states in design, this research focuses on the ultimate

limit-state design of CFS members.

2.2.1 The Ultimate Limit-State Equation

Equation 2.19 represents a favourable ultimate limit-state. This is where the structural resistance R exceeds the load effect E.

𝐸 < 𝑅 (2.19)

If the limit state represented by Equation 2.19 is not satisfied, it is assumed that structural failure occurs. An unambiguous distinction is made between a desirable and

undesirable state by representing Equation 2.19 in the form expressed as the

fundamental form of the limit-state equation in Equation 2.20.

𝑅 − 𝐸 = 0 (2.20)

The ultimate limit-state is represented by G. The safe-state of the limit-state is where the

mathematical difference between the structural resistance and the load effect must be greater or equal to zero. This is expressed in Equation 2.21.


19

𝐺 = 𝑅 − 𝐸 ≥ 0 (2.21)

The structural resistance and the load effect can be represented by functions of i and j number of X variables respectively. Randomness of the limit-state equation is the effect

of these X variables. The variables R and E are expressed in the form of Equation 2.22 and 2.23 respectively.

𝑅 = 𝑓 𝑋��, 𝑋�{, 𝑋��, … , 𝑋�� (2.22)

𝐸 = 𝑓 𝑋��, 𝑋�{, 𝑋��, … , 𝑋�� (2.23)

Uncertainties in the load effect and the structural resistance implies that they are both

random variables. To get a definitive mathematical difference between the two is therefore problematic. Variability in the load effect and structural resistance are

important things to consider when applying a structural reliability analysis (Nowak and

Collins, 2000).

The load effect generally consists of long and short-term loads. Although the prediction

methods of permanent loads and other long-term loads are fairly accurate, there is still

an underlying level of uncertainty that needs to be accounted for. For example, the density of a material cannot be exactly predicted or consistent throughout the structure.

Moreover, finding the exact force of imposed loads, wind loads, and other temporary loads

are near to impossible to predict at any given time.

Structural resistance generally depends on the yield stress of the material and member

geometry. It is unrealistic to suggest that these two properties are consistent along the

length of a member. The yield stress of a material is highly dependent on a consistent chemical composition. The resulting thickness of the member is as accurate as the method

of manufacture.

2.2.1.1 Probability of Failure Because the load effect and the structural resistance are both random variables in most cases, the validity of Equation 2.21 is not absolutely guaranteed. Thus, it can be said that

the limit-state equation expressed in Equation 2.21 can be exceeded and that there exists

a probability of failure. In terms of Equation 2.19, the probability of failure denoted by pf of the limit-state is expressed in Equation 2.24.

𝑝� = 𝑃 𝐸 > 𝑅 (2.24)


20

As the load effect and the structural resistance are random variables in most cases, they can be expressed as probability distribution functions. The probability distributions for

the load effect and the structural resistance can be obtained once their respective random

variables have been defined. Each probability distribution function can be described by a

mean μ, a standard deviation σ, a skewness α, and a kurtosis ε. These statistical

parameters, referred to as statistical moment parameters, are important characteristics in determining the probability of failure. They influence the relative positioning,

dispersion, and shape of the two probability density functions, as well as their relative

distances from one another.

Generally, the statistical moment parameters of each probability distribution function describe how the probability of failure will be affected. The relative position of the two

curves is affected by the means of each curve. The probability of failure increases as the

distance between the two curves decrease. The dispersion of the two curves is affected by the standard deviation of each curve. The probability of failure decreases the narrower

the curves become. If any of the probability distribution functions take on an un-

symmetrical shape, the skewness of the curve effects the shape of the curves. Although not exclusively descriptive of a curve shape, the skewness of a probability distribution

function is still significant in determining the probability of failure. The shape of the

probability distribution function may affect the overlapping area between the two probability distribution functions, thus increasing or decreasing the probability of failure.

The probability density functions are shown graphically in Figure 2.8 by Bauer (2016).

Observing Figure 2.8, it is evident that, depending on the variability of their respective random variables, a broad range of realisations can be observed by either the load effect

or the structural resistance. Additionally, there exist certain realisation combinations

where the limit-state equation is not satisfied.


21

Figure 2.8: Probability density functions of load effect and structural resistance (Bauer, 2016)

2.2.1.2 Measure of Structural Reliability To help describe the procedure, an initial assumption is made for simplicity. It is assumed

that that the probability density functions of the structural resistance and the load effect

both have symmetrical normal distributions with different statistical moment parameters. It is noted that this is unlikely in application and this assumption is being

made purely for the purposes of description.

From, Equation 2.21, the safe-state of the limit-state equation represented by G can be

defined as its own random variable. For the normally distributed functions E and R, statistical moment parameters of µG, σG and αG of the random variable G are given in

Equation 2.25, 2.26 and 2.27 respectively (Holický, 2009).

𝜇¡ = 𝜇� − 𝜇� (2.25)

𝜎¡ = 𝜎�{ + 𝜎�{ + 2 ∙ 𝜌�� ∙ 𝜎�{ ∙ 𝜎�{ (2.26)

𝛼¡ =𝜎�� ∙ 𝛼� − 𝜎�� ∙ 𝛼�

𝜎�{+𝜎�{ � { (2.27)

ρRE is the correlation coefficient of the structural resistance and the load effect. Often the

structural resistance and the load effect are independent of one another, so that ρRE = 0

(Holický, 2009).

Prob

abili

ty D

ensi

ty φ

Structural Resistance R

Magnitude of the Load Effect E and Resistance R

Load Effect E


22

Figure 2.9 shows the probability density function of the limit-state represented by G. In this figure by Bauer (2016), it is observed that a portion of the curve extends below zero.

With the curve encapsulating all possible realisations of G, the realisations of the limit-

state that extend below zero are the realisations that do not satisfy Equation 2.21. Therefore, it can be said that the area under this part of the curve is the probability of

failure, represented by the shaded area in Figure 2.9.

With reference to Equation 2.24, the probability of failure can be statistically expressed

in the form of Equation 2.28. This equation expresses G as a statistical function of R and E. The symbol ΦG is the cumulative probability distribution function of the limit-state G.

The probability of failure is therefore the area under the limit-state probability

distribution function where the limit-state G is below zero.

𝑝� = 𝑃 𝑅 − 𝐸 < 0 (2.28a)

= 𝑃 𝐺 𝑅, 𝐸 < 0 (2.28b)

= 𝛷¡ 0 (2.28c)

Figure 2.9: Probability density function of the limit-state G (Bauer, 2016)

The reliability margin, denoted by the symbol β, can be defined as the number of standard

deviations the mean of the limit-state function G is from zero, given that G is a normally

distributed function (Holický, 2009). This implies that β is dimensionless. The reliability

margin and the probability of failure are therefore related, as the reliability margin

Limit-State G

Magnitude of Limit-State G

Prob

abili

ty D

ensi

ty φ

0


23

increases with a decrease of probability of failure. The relation is expressed in Equation 2.29 (Nowak & Collins, 2000).

𝑝� = 𝛷¦ −𝛽 𝑜𝑟𝛽 = −𝛷¦§� 𝑝� (2.29)

ΦU is the standardised normal distribution function.

The value of β can be calculated with the moment statistical parameters of the load effect

and the structural resistance. β is the ratio of the mean of the limit-state function to the

standard deviation of the limit-state function. With reference to Equation 2.25 and 2.26,

β is expressed in Equation 2.30.

𝛽 = 𝜇¡𝜎¡

=𝜇� − 𝜇�

𝜎�{ + 𝜎�{ + 2 ∙ 𝜌�� ∙ 𝜎�{ ∙ 𝜎�{(2.30)

For general reliability verification procedures, SANS 10160-1 (2011) suggests a reliability class 2 to be used. The minimum level of reliability for a reliability class 2 structure is

βt = 3. The basis of this thesis assesses whether the inherent reliability of the DSM

equations for compression members formulated in SANS 10162-2 (2011) achieve the

minimum level of reliability prescribed by SANS 10160-1 (2011).

2.2.1.3 Space of State Variables Supplementary to the reliability analysis, the space of state variables may be described. The state variables considered are the structural resistance and the load effect. The space

of state variables is a two-dimensional space where the limit-state function separates the

safe domain and the failure domain. This is depicted in Figure 2.10 by Nowak and Collins (2000). The load effect is plotted on the x-axis and the structural resistance is plotted on

the y-axis.


24

Figure 2.10: Two-dimensional space of state variables (Nowak and Collins, 2000)

Since the load effect and the structural resistance are random variables, a joint density function can be defined. As in Figure 2.10, the joint density function is separated into the

safe domain and the failure domain, separated by the limit-state function. This is depicted

as a 3-dimentional plot in Figure 2.11 (Nowak and Collins, 2000).

Figure 2.11: 3-Dimentional plot of the joint density function (Nowak and Collins, 2000)

Limit-State Function G = 0 (R = E)

Failure domain (R < E)

Safe domain (R > E)

45°

E

R

µE

µR


25

Further, the space of state variables can be visualised in a standardised space to show the reliability margin. This is depicted in Figure 2.12 by Haldar and Mahadevan (2000). As

the limit-state equation G is not necessarily linear, the design point (r0d, e0d) must be

determined. The design point is a point on the limit-state equation that is closest to the origin of a standardised space. The distance that the design point is from the origin of the

standardised space is measured as the reliability margin β. It is possible for multiple

design points to exist if the limit-state equation is non-linear. Figure 2.12 shows an example of the reliability margin for a non-linear limit-state function

Additional to the reliability analysis is the directional vector α of the reliability margin β.

Sharing the same symbol as the skewness of a distribution function, one must not confuse

thee two as they have completely different meanings. Also referred to as the sensitivity

factors, the directional vector is split into two components relating to the load effect αE

and the structural resistanceαR. Shown in Equation 2.31 and 2.32, it is evident that the

sensitivity factors for the load effect and the structural resistance are directional cosines of the reliability margin from the transformation to a standardised space. These are

normal to the failure boundary (Holický, 2009).

𝛼� = −𝜎�

𝜎�{ + 𝜎�{(2.31)

𝛼� =𝜎�

𝜎�{ + 𝜎�{(2.32)

Figure 2.12: Reliability margin and sensitivity factors in a standardised space for a non-linear limit-state equation (Haldar and Mahadevan, 2000)


26

A simplified statistical approach in determining the reliability margin and the associated sensitivity factors has been presented in this section. Although the above explanation

expressed the limit-state equation as a function of two variables, it is possible for the

limit-state equation to be expressed as a function of more than two variables. This implies that the above procedure is not always possible to follow. Thus, in a more realistic

multivariate case where there are n-number of random variables, a different form

procedure is required (Nowak and Collins, 2000). A procedure in determining the reliability for such complex situations is subsequently discussed.

2.2.2 Reliability Simulation Methods

There exist many methods of reliability analysis for complex situations. The following

reliability simulations were discussed as the most appropriate for the investigation of this thesis.

2.2.2.1 First Order Reliability Method The First Order Reliability Method (from here on referred to as FORM) is a basic, yet

efficient reliability method. Holický (2009) states that the EN 1990 design values are

based on the FORM reliability method.

Initially developed by Hasofer and Lind, the FORM analysis aims to reduce a multivariate limit-state case to a normally distributed variable problem (Breitung, 2015). This

procedure determines equivalent normal distribution statistical parameters of mean µe

and standard deviation σe at each design point, and iterates until convergence is reached

at the design point corresponding to the smaller βvalue.

According to Holický (2009), the transformation of the non-normally distributed case to

the normally distributed case is based on two conditions. The first is that the cumulative distribution functions and the equivalent standardised normal distribution variables

should be equal. The second is that the probability density functions of the n-number of

random variables should be equal to that of the equivalent normal density functions. Both of which must occur at the design point denoted as x*. These conditions are described in

Equation 2.33 and 2.34 respectively (Holický 2009).


27

𝛷¨ 𝑥∗ = 𝛷¦𝑥∗ − 𝜇¨m

𝜎¨m(2.33)

𝜑¨ 𝑥∗ =1𝜎¨m

𝜑¦𝑥∗ − 𝜇¨m

𝜎¨m(2.34)

Subsequently, the mean and the standard deviation of the equivalent normal distribution of variable X are determined in Equation 2.35 and 2.36 respectively.

𝜇¨m = 𝑥∗ − 𝜎¨m ∙ 𝛷¦§� 𝛷¨ 𝑥∗ (2.35)

𝜎¨m =1

𝜑¨ 𝑥∗𝜑¦

𝑥∗ − 𝜇¨m

𝜎¨m=

1𝜑¨ 𝑥∗

𝜑¦ 𝛷¦§� 𝛷¨ 𝑥∗ (2.36)

The process to solve the reliability problem is an iterative process, since the limit-state

equation is, in general, non-linear. The FORM analysis attempts to generate a tangent to the non-linear limit-state equation. With enough iterations, the generated tangent runs

through the design point on the non-linear limit-state function. The distance

perpendicular to the tangent from the design point to the origin of the standardised space is therefore the shortest. The number of iterations influence the accuracy of the outcome.

With reference to Figure 2.10 and Figure 2.12, this is shown in Figure 2.13 (Holický,

2009).

Figure 2.13: First order reliability method (Holický, 2009)

𝑈� =𝑅 − 𝜇�𝜎�

𝑈� =𝐸 − 𝜇�𝜎�

Non-linear Limit-state function

Design Point x*(r0d, e0d) Tangent

generated by FORM

β


28

2.2.2.1.1 First Order Reliability Method Calculation Procedure A summary of the iterative FORM procedure is presented in the following ten steps (Holicky, 2009):

1. Formulated as 𝐺 𝑋 = 0 , the limit-state function is expressed in terms of

theoretical models of basic variables 𝑋 = 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬

2. An initial design point 𝑥∗ = 𝑥�∗, 𝑥{∗, 𝑥�∗, … , 𝑥¬∗ is decided and assessed.

3. Equivalent normal distributions are found at the design point for all basic

variables. This is done by using Equations 2.35 and 2.36.4. From the standardised random variables 𝑈 = 𝑈�, 𝑈{, 𝑈�, … , 𝑈¬

corresponding to the design point, the transformed standardised design point

𝑢∗ = 𝑢�∗, 𝑢{∗, 𝑢�∗, … , 𝑢¬∗ is determined using Equation 2.37.

𝑢�∗ =𝑥�∗ − 𝜇¨®

m

𝜎¨®m (2.37)

5. Denoted as {D}, the vector of the partial derivatives for the limit-state function

is evaluated at the design point. This is done in respect of the standardised

variables and is shown in Equation 2.38.

𝐷 =

𝐷�𝐷{⋮𝐷¬

𝑤ℎ𝑒𝑟𝑒𝐷� =𝜕𝐺𝜕𝑈�

=𝜕𝐺𝜕𝑋�

∙𝜕𝑋�𝑈�

=𝜕𝐺𝜕𝑋�

∙ 𝜎¨®m (2.38)

6. From this, an estimation of reliability index β is calculated using Equation

2.39.

𝛽 = −𝐷 ± 𝑢∗

𝐷 ± 𝐷𝑤ℎ𝑒𝑟𝑒 𝑢∗ =

𝑢�∗𝑢{∗⋮𝑢¬∗

(2.39)

7. With reference to Equations 2.31 and 2.32, the sensitivity factors for each

variable are then determined using Equation 2.40.

𝛼 =𝐷

𝐷 ± 𝐷(2.40)


29

8. A new design point is updated for n – 1 standardised and original basicvariables by using Equation 2.41 and 2.42 respectively.

𝑢�∗ = 𝛼�𝛽� (2.41)

𝑥�∗ = 𝜇¨®m − 𝑢�∗𝜎¨®

m (2.42)

9. From the limit-state function 𝐺 𝑥∗ = 0, the design value of the remaining n-

number of variables is recalculated.10. Steps 3 to 9 are iteratively repeated until the reliability index converges to the

design point with the required accuracy.

Depicted by Lopez and Beck (2012), Figure 2.14 shows a 3-dimentional graphical

interpretation of the FORM analysis. It is to be noted that it regards the area under the hat-shaped curve that is past the tangent generated by the FORM analysis as the

probability of failure. However, as the limit-state function in Figure 2.14 is non-linear and

convex, the FORM analysis perceives a part of the area as unsafe, when in fact it is safe. This implies that the FORM analysis is slightly conservative, given that the limit-state

function is convex as depicted in Figure 2.14. The closer the limit-state function is to a

linear function, the less conservative it becomes.

Figure 2.14: Graphical representation of the FORM analysis (Lopez and Beck, 2012)

FORM Tangent

β

UE UR


30

For the case where the limit-state function is concave, the reliability margin that the FORM analysis determines may be under-conservative. This could prove to be

problematic as the reliability margin generated by the FORM analysis may be higher than

the true level of reliability. The Monte Carlo Simulation (from here on referred to as MCS) better predicts the level of reliability, provided that enough samples are generated.

Comparing the results of the MCS and the FORM analysis, the general shape of the limit-

state function could be estimated.

2.2.2.2 Monte Carlo Simulation Method Cardoso et al. (2008) classifies three levels of reliability methods. Each level varies in accuracy and ease of calculation. Level 1 methods or semi-probabilistic methods

approximates the probability of failure using partial factors. This method is most used in

practice as it requires little calculation effort. However, the calibration of the partial factors in level 1 methods is based on level 2 or level 3 methods to achieve consistent

performance. Level 2 methods, or approximate probabilistic methods, include the FORM

analysis. This method is more accurate than the level 1 methods but require moderate computing efforts. Level 3 methods, or exact probabilistic methods, are where the

probability of failure is determined from the joint probability of the random variables.

Although these methods are the most accurate, they require the highest level of computing effort. The MCS is an example of a level 3 reliability method.

As accurate as the MCS may be, the method is not wide-spread in the analysis of

structural reliability. This is because it is not as computationally efficient as the more common level 2 methods. For structural reliability applications, the tail-end of

performance is evaluated, requiring large numbers of MCS samples to adequate accuracy.

The use of the MCS in this investigation is purely to determine whether the shape of the limit-state function is concave or convex. Additionally, unlike the MCS analysis, the

FORM analysis easily yields results for the sensitivity factors.

2.2.2.2.1 Monte Carlo Simulation Calculation Procedure With reference to Equation 2.28, a brief description of the MCS analysis for structural

reliability is presented. It is to be understood that the probability of failure is expressed in Equation 2.43 (Cardoso et al., 2008). X1, X2, X3, …, Xn are the random variables

associated with the limit-state. The term 𝑔 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ ≤ 0 is an expression of the

violation of the limit-state equation. The term 𝑓 ²,¨u,¨³,…,¨´ 𝑥�, 𝑥{, 𝑥�, … , 𝑥¬ is the joint

probability density function of realisations of all considered random variables.


31

𝑝� = 𝑃 𝑔 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ ≤ 0 (2.43a)

= ⋯¶·�

𝑓 ²,¨u,¨³,…,¨´ 𝑥�, 𝑥{, 𝑥�, … , 𝑥¬ 𝑑𝑥�𝑑𝑥{𝑑𝑥�, … , 𝑑𝑥¬ (2.43b)

A descriptive interpretation of Equation 2.43 is that the area under the joint probability

density function where the limit-state function is equal to or less than zero is the

probability of failure. Depending on how many iterations are run, the MCS determines the probability of failure as expressed in Equation 2.44.

𝑝� =1𝑁

𝐼 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬

¸

�¹�

(2.44)

𝐼 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ is a function that is defined by Equation 2.45.

𝐼 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ = 1 𝑓𝑜𝑟𝑔 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ ≤ 0 (2.45a)

𝐼 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ = 0 𝑓𝑜𝑟𝑔 𝑋�, 𝑋{, 𝑋�, … , 𝑋¬ > 0 (2.45b)

The symbol N is the number of independent sets of values 𝑥�, 𝑥{, 𝑥�, … , 𝑥¬ obtained based

on the probability distribution function of each random variable. Using Equation 2.29, the

reliability level can be calculated once the probability of failure is determined.

Achintya & Mahadevan (2000) states that the accuracy of the probability of failure is

dependent on the number of iterations. With more MC iterations, the probability of failure would approach the true value. A method to evaluate the accuracy of the MC results is to

assess the coefficient of variation of the estimated probability of failure. This is depicted in Equation 2.46 (Achintya & Mahadevan, 2000).

𝐶𝑂𝑉 =

1 − 𝑝� ∙ 𝑝�𝑁

𝑝�

(2.46)

pf is the probability of failure estimated from the MC simulation and N is the number of trials conducted. For the purposes of this study, the probability of failure relates to a

reliability level of β = 3 and 10 million trials were conducted for each MC simulation. The

resulting coefficient of variance was 0.86%.

The point of departure of the discussion of the MCS is to convey the fact that, although it

requires a greater computing effort, the MCS yields more accurate βvalues than the


32

FORM analysis. In this study, all βvalues presented in the results are from the FORM

analysis. However, for each of the considered load combinations, a MCS was performed to estimate the shape of the limit-state function. If the βvalue determined from the MCS

was greater than the βvalue from the FORM analysis, the limit-state function is convex,

and the FORM analysis is conservative. If the βvalue determined from the MCS was less

than the βvalue from the FORM analysis, the limit-state function is concave, and the

FORM analysis is under-conservative.


33

2.3 Model Factor The model factor or model uncertainty relates the test results to the predicted results of a model (Holický et al., 2015). The model factor quantifies the epistemic level of

uncertainty present in a prediction model. As in all models, it is rare for a prediction model

to exactly predict test results. Sources of uncertainty stem from the inherent variability of materials, loads and member geometry, as well as epistemic uncertainty due to

imperfect models with simplifications and statistical uncertainty. The model factor

therefore plays a significant role in the reliability analysis. To account for the sources of uncertainty, the model factor for the DSM was introduced in this study.

Expresses in Equation 2.47, the model factor is the ratio of the test result to the predicted

result of a prediction model. Model factors exist for the load effect and the structural

resistance. The model factors associated with the load effect are dependent on the type of load. Each load type has a different model factor, as the difficulty of data collection and

analysis may vary for the different types of loads. The model factors associated with the

structural resistance are dependent on the level of approximation of the prediction model including test aspects such as section shape and buckling mode. The reliability margin

investigated in this study covers the design procedure for the buckling modes of failure as

discussed in Section 2.1.2.2.1. Therefore, a separate resistance model factor is used for each buckling mode.

𝜕 = 𝐹½m¾½𝐹¿~kml

(2.47)

It is to be noted that Equation 2.47 is the definition of a model factor for a single test. The

model factors used in this study are considered as probability distribution functions,

derived from a historic database of tests found in literature by Ganesan and Moen (2010). This is to say that the model factor statistically describes the predictive ability of the DSM

model. Error! Reference source not found. shows the test-to-predicted ratios for all the

CFS columns considered in the test database of Ganesan and Moen (2010) for the DSM. The test-to-pedicted ratios are expressed as a function of global slenderness. Of the test

database, 245 were of CFS lipped C members.


34

Figure 2.15: Test-to-predicted ratios of CFS columns as a function of global slenderness for the DSM (Ganesan and Moen, 2010)

Holický et al. (2015) categorises the effect that the model factor has on the level of uncertainty into three groups. The groups are dependent on the value of the sensitivity

factors of the model factors obtained from a FORM analysis. Table 2.2 shows the

characterisation of the model factors.

Table 2.2: Categorisation of the model factor (Holicky, et al., 2015)

Grouping FORM Sensitivity Value

1 – Minor Effect 𝛼À < 0.32

2 – Significant Effect 0.32 < 𝛼À < 0.80

3 – Dominating Effect 0.80 < 𝛼À

𝑃 ½m¾½𝑃 ¬Á

Lip C Lip Z Plain Z Plain C Hats Ptest/Pn = 1

𝜆] = Â𝑃_ 𝑃]^mÃ Ä

�.Å


35

Holický, et al. (2015) states that by comparing physical test results with prediction model results, model uncertainty is obtained. Additionally, structural conditions should also be

considered if needed. This concept is depicted in Figure 2.16. The significant factors that

affect the test results, model results and structural-specific conditions are directly dependant on how the failure modes of a structural member are analysed. The following

three aspects should be considered when quantifying model uncertainty (Holický, et al., 2015).

1. Test results: well-calibrated test methods are typically unbiased with a test uncertainty mean of unity. The variability depends on external factors such as the

skill level of staff and the type of tests conducted. 2. Model results: simplifications and other relevant computational options for

numerical computer models should be reflected. Uncertainties in the input data

should rather be reflected in the description of the basic variables. 3. Structure-specific conditions: Depending on the application of the structural

member, it may be more desirable to quantify the differences between structure

and specimens rather than to investigate the test uncertainty. Characters of thetest uncertainty may represent the differences between structure and specimens.

The model factors used in the tests conducted in this thesis were quantitative

representations only of the differences between the test results and the model results,

which will include items 1 and 2 above. Structure-specific conditions were not considered.


36

Figure 2.16: General concept of the model factor (Holický et al, 2015)

1. Test Results- Uncertainty of test method (accuracy of gauges, errors in readings, definition of ultimate resistance, etc) - Uncertainty in execution of an individual specimen/test (differences in strengths in the test specimen and control specimen, differences in actual dimensions and those measured or prescribed) - Other effects (not covered by tests such as time-variant effects)

2. Model Results- Model simplifications (assumed stress distributions, boundary conditions) - Description of input data (assumptions concerning variables with unknown values, tensile strength, fracture energy, plastic strain) - Computational options (discretisation, type of finite elements, boundary conditions – simplifications made by analysts)

Comparison

Observed uncertainty

MODEL UNCERTAINTY

Test uncertainty

3. Structure-specific conditions- Production quality and control of execution - Boundary conditions (supports, continuous members, integral structures) - Loading conditions (transfer, combination of shear and bending moments) - Thermal and moisture conditions - Size effect


37

3 CHAPTER 3: Assessment of Cold-Formed Steel

Columns Designed with the Direct Strength Method

This chapter covers the processes that were followed to determine the reliability level of

DSM designed CFS columns using the FORM analysis. To do this, a representative

compression member was chosen. Load conditions were parametrically varied, and several member lengths were considered. To proceed with the DSM, the use of the CUFSM

computer program, which incorporates the FSM, was required to distinguish the buckling

modes and at which member lengths the different buckling modes govern design resistance.

3.1 Basis of Reliability Analysis The basis of this study was to conduct a reliability analysis on a suitable limit-state

equation formulation. Based on published literature and accepted published models, the random variables of the load effect and the structural resistance were chosen to represent

a typical compression element in a structure.

The chosen member conforms to the prequalification criterion of SANS 10162-2 (2011) as

summarised in Table 2.1. The member has two characteristic properties that describe the behaviour of the member under a given load. These properties are the material and the

geometric properties. Each of these properties are to be considered as continuous random

variables when conducting a reliability analysis. Typically, the yield stress of the material represents the material properties and the thickness of the member represents the

geometric properties. However, based on research conducted by Bauer (2016), it was found that, when compared to the thickness and the yield stress, the resistance model factor

governs uncertainty of the structural resistance. This is shown in Figure 3.1. Therefore,

the thickness and the yield stress were taken as deterministic values.


38

Figure 3.1: Sensitivity factors of structural resistance random variables (Bauer, 2016)

To proceed with the reliability analysis, however, the limit-state function was described

in terms of the semi-probabilistic and full probabilistic formulations. The semi-probabilistic formulation of the limit-state equation is a representation of the limit-state

in terms of codified values for the load effect and the structural resistance. To evaluate

the inherent level of reliability achieved by the codified semi-probabilistic formulation of the limit-state, a full probabilistic formulation of the limit state is to be assessed. Here,

the load effect and the structural resistance are represented by associated random

variables. Each of the considered random variables are expressed as probability distribution functions.

This study assesses the inherent reliability level implied by the semi-probabilistic

formulation of the limit-state. Specifically, the design structural resistance is based on the SANS 10162-2 (2011) formulation of the DSM using a prequalified column section.

According to the DSM, the structural resistance is dependent on the buckling mode. In

this study, the member length and profile varied to induce all the buckling modes. A range of load ratios were considered, based on SANS 10160-1 (2011).

3.1.1 Semi-Probabilistic Formulation of the Limit-State

The semi-probabilistic formulation of the limit-state expresses the design load and the

design structural resistance as characteristic values multiplied by partial factors.


39

Therefore, the load effect and the structural resistance are deterministic values. The semi-probabilistic limit-state requirement is expressed in Equation 3.1.

𝑅k − 𝐸k ≥ 0 (3.1)

Rd is the codified design value for structural resistance and Ed is the codified design value

of the load effect.

For the purposes of this study, the codified design value of the structural resistance prescribes to the DSM formulation presented in SANS 10162-2 (2011). This is to say that

the design value of the structural resistance Rd in the semi-probabilistic formulation of

the limit-state will equal the design compressive critical buckling load Nd.

The codified design value of the load prescribes to SANS 10160-1 (2011) for the various loads. For the purposes of this study, the considered load types were permanent, imposed

and wind loads. The design values for each of the loads are represented in terms of their respective partial factors, combination factors and characteristic values. The partial and

combination factors are dependent on the considered load combinations. The

characteristic values of the loads were used as inputs to the full probabilistic formulation of the limit-state.

To obtain the design load of the semi-probabilistic formulation of the limit-state, the limit

state requirement of Equation 3.1 was equated to zero. Hence, the total design load effect

equated to the design value of the structural resistance.

3.1.2 Full Probabilistic Formulation of the Limit-State

To present the full probabilistic formulation of the limit-state, the basic variables of the limit-state were expressed as probability distribution functions. The statistical moment

parameters of the basic random variables are expressed in terms of the characteristic values of the loads. The characteristic values of the loads were obtained from the semi-

probabilistic formulation of the limit-state.

The full probabilistic formulation of the limit state can be expressed in Equation 2.20,

where R and E are functions of associated random variables.

The structural resistance of the member in the full probabilistic formulation of the limit-state was calculated using the DSM described in Section 2.1.2.2.3. However, to attain a

true, unconservative representation of the structural resistance, mean values of the

material and geometric properties were used rather than the characteristic values.


40

Additionally, no partial factors or reduction factors were used in the calculation. The mean thickness was taken as the thickness given in the specification of the cross-sectional

properties of the member. The overall structural resistance of the member in the full

probabilistic formulation of the limit-state is represented by the symbol R. In this study, the structural resistance is comprised of the mean compressive critical buckling load Nµ

and the structural resistance model factor 𝜕R.

The associated random variables are discussed and explained in subsequent sections.

Essentially, the full probabilistic formulation of the limit-state was analysed to evaluate the inherent level of reliability of the semi-probabilistic formulation of the limit-state

presented in SANS 10162-2 (2011).

3.1.3 Process of Reliability Analysis

The process of the reliability analysis is shown graphically in Figure 3.2. The geometric

and material properties of a chosen compression member were initially defined. Using the CUFSM computer program, the design and mean yield capacities, represented by the

symbols Ny,d and Ny,µ respectively, were obtained. Subsequently, signature curves were

generated for the design and mean capacities. For each signature curve, the buckling modes were identified in terms of their respective buckling half-wavelengths and load

factors.

Using the design equations of the SANS 10162-2 (2011) formulation of the DSM, the semi-probabilistic formulation of the limit-state equation was established. Here, the design

compressive capacity of the member Nd was calculated. The characteristic values of the

loads were then determined from Nd, since Equation 3.1 was used and equated to zero. The codified combination factors and partial factors are dependent on the ratio of loads

and the combination of these loads.

A FORM analysis of the full probabilistic limit-state equation was performed on the full

probabilistic formulation of the limit-state. Each of the variables in the limit-state are expressed in terms of associated basic variables. The statistical parameters of the basic

random variables of the load effect are based on the characteristic values of the considered

loads. These were obtained from the semi-probabilistic formulation of the limit-state. The structural resistance considered the mean compressive capacity of the member Nµ.

Model factors for the load effect and the structural resistance were included in the full

probabilistic formulation of the limit-state. They were considered as continuous random


41

variables. The statistical parameters of the model factors were based on literature and historical tests.

The results of the full probabilistic formulation of the limit-state presented the reliability

levels at the different load combinations for each buckling mode. At the corresponding

load combinations for which the reliability level was determined, the sensitivity factors were also determined. Sensitivity factors reflect the influence each of the respective

variables in the limit-state equation on the reliability level.


42

Figure 3.2: Flow diagram depicting the general procedure of the reliability analysis

The remainder of the chapter is to be read in conjunction with observing Figure 3.2.

Herewith following, all steps shown in Figure 3.2 are explained as sub-headings in detail.

Geometric Properties

Start

Input

Material Properties

Properties Obtain Yield Capacities

(Ny,d & Ny,µ)

Obtain Signature Curves

Identify Buckling Mode Capacities

Local

Global

Limit-State: Semi-Probabilistic Formulation (Nd)

Rd–Ed=0

Obtain characteristic loads from Nd for

range of load ratios and load

combinations

R–E=0

Where𝑅 = 𝜕� ∙ 𝑁Ç

and𝐸 = 𝐺 + 𝜕È𝑄 + 𝑊 𝛽 = −{𝐷}±{𝑢∗}

Ë{𝐷}±{𝐷}

𝛼 = {𝐷}

Ë{𝐷}±{𝐷}

End

FSM

DSM

Limit-State: Full Probabilistic Formulation (Nµ)

Distortional

FORM

Mean value of material yield stress

(fy,µ)

Design value of material yield stress

(fy,d)

Where Rd=Nd

43

3.2 Analysis of Cold-Formed Steel Compression Member The section geometry and profile of the chosen CFS compression member was obtained by

a previous study conducted by Bauer (2016). In the study, the plain lipped C-section was chosen, shown in Figure 3.3. This is because it is considered as a common compression

element in South African CFS design.

Furthermore, as only the local and global buckling modes were induced for the considered

section, alterations were made to the section to induce distortional buckling. It was established that by adding a stiffener to the web of the member, the distortional and global

buckling modes were induced. This is shown in Figure 3.4. Therefore, the local and global

buckling analyses were performed on a plain lipped C-section and the distortional and global buckling analyses were performed on a stiffened lipped C-section with similar

geometric and sectional properties.

3.2.1 Member Geometric Properties

The geometrical properties of the chosen plain lipped C-section and the stiffened lipped C-Section are shown in Table 3.1. The CUFSM computer program required the centreline

dimensions as input parameters for modelling. These geometric properties of the sections

were used for the calculation of Nd for the semi-probabilistic formulation of the limit-state and Nµ for the full probabilistic formulation of the limit-state. Figure 3.3 and Figure 3.4

show the geometric detailing of the plain lipped C-section and the stiffened lipped C-

section respectively.


44

Figure 3.3: Plain lipped C-section with centreline dimensions

Figure 3.4: Stiffened lipped C-section with centreline dimensions

h

w

l r

h

w

l r

2s

s


45

Table 3.1: Geometrical properties of compression elements

Plain Lipped C-Section Stiffened Lipped C-Section

Section Property Symbol Dimension (mm)

Centreline Dimension (mm)

Dimension (mm)

Centreline Dimension (mm)

Web height h 89 83.5 82.373 76.875

Flange width w 41 35.5 41 35.5

Lip height l 10.1 7.35 10.1 7.35

Angle radius r 2 2.375 2 2.375

Thickness t 0.75 0.75 0.75 0.75

Member length L 4000 4000 4000 4000

Stiffener length s 0 0 8.75 8

The sectional properties required to perform the calculations in Section 2.1.2.2.3 were obtained from the CUFSM computer program. For the cross-sections in Figure 3.3 and

Figure 3.4, the section properties that were used in the codified DSM calculations are

shown in Table 3.2. Observing Table 2.1, the analysed sections satisfy the prequalification limitations for the DSM design method.


46

Table 3.2: Sectional properties obtained from CUFSM

Plain Lipped

C-Section

Stiffened Lipped

C-Section

Geometric

Property

Unit Value Value

A mm2 138.0202 138.0213

J mm4 25.8788 25.879

xcg mm 12.6011 12.5666

ycg mm 44.125 44.1247

Ixx mm4 176897.216 154025.7559

Iyy mm4 31731.8714 28810.2526

xo mm -18.9829 -17.0042

yo mm 44.125 44.1247

Iw mm6 51121114.09 43982030.78

A is the gross area of the cross-section, J is the torsion constant for the cross-section, xcg

and ycg are the coordinates of the centre of gravity of the section, Ixx and Iyy are the moments of inertia about the x- and y-axes respectively, xo and yo are the coordinates of

the shear centre and Iw is the warping constant for the cross-section.

It is to be noted that the stiffened lipped C-section member shares the same cross-

sectional area as the plain lipped C-section member. This is to say that the same amount of steel was used for each section.

3.2.2 Member Material Properties

The material properties of the members required for input values of the CUFSM computer

program included the modulus of elasticity, Poisson’s ratio, the shear modulus of elasticity and the yield stress. The same material properties were taken for the plain and lipped C-

sections respectively. The material properties of grade G550 steel were used. These values

are shown in Table 3.3.


47

Table 3.3: Material properties of Grade G550 high strength steel

Material Property Symbol Value Unit

Modulus of elasticity E 203,000 MPa

Shear modulus G 78,077 MPa

Poisson’s ratio ν 0.3 N/A

Grade G550 steel conforms to SANS 4998: Continuous hot-dip zinc-coated carbon steel

sheet of structural quality (from here on referred to SANS 4998). As the yield stress of the steel was 550MPa, this conforms to the DSM pre-qualification limitations presented in

Table 2.1 for the plain and stiffened lipped C-sections.

Clause 1.5.1.4(b) of SANS 10162-2 (2011) limits the yield stress of the steel based on the

thickness of the element. The clause states that if the thickness of an element is between 0.6mm and 0.9mm, it should be assumed that the yield stress used in calculating Nd is

taken as the lessor of 495MPa or 90% of the yield stress. The limitation of the yield stress

was applied only to the calculation of Nd and not of Nµ. This is because, as explained in Section 3.1.2, the mean compressive capacity of the column member is to be a true and

unconservative representation of the steel resistance.

3.2.3 Design and Mean Yield Loads

The design and mean yield loads for the sections were calculated by multiplying the yield stress with the gross cross-sectional area of each section. The gross sectional areas used

to calculate Nd and Nµ are equivalent, however the yield stresses used to calculate Nd and

Nµ are different. This is because the yield stress to calculate Nd is the characteristic yield stress and the yield stress used to calculate Nµ is the mean yield stress.

The calculation of the mean yield stress was determined by using Equation 3.2 from

Holický (2009).

𝜇�Ì = 𝑓_ + 2 ∙ 𝜎�Ì (3.2)

where

𝜎�Ì = 0.1 ∙ 𝜇�Ì (3.3)


48

Observing Equations 3.2 and 3.3, the mean value and the standard deviation of the yield stress are co-dependant. Therefore, an iterative procedure is required to ensure

convergence of the moment parameters of the yield stress. The design yield stress was

assumed to be 495MPa due to the limiting clause 1.5.1.4(b) of SANS 10162-2 (2011). Satisfying Equations 3.2 and 3.3, the mean yield stress was calculated as 687.5MPa. The

yield stresses were then multiplied by the cross-sectional area of the sections, found in

Table 3.2. This resulted in mean and design yield loads as shown in Table 3.4.

Table 3.4: Yield stresses and yield capacities for the plain and stiffened lipped C-sections

Unit Plain Lipped

C-Section

Stiffened Lipped

C-Section

fy,d MPa 495 495

fy,µ MPa 687.5 687.5

Ny,d N 68,320 68,320

Ny,µ N 94,889 94,889

The study by Van Wyk (2014) investigated the tear-out resistance of screwed connections

in CFS members. In the study, it was found that the mean yield stress of G550 grade CFS

was 657.7MPa and the standard deviation was 70.5MPa. Therefore, by comparing the assumed value of the mean yield stress used in this study to that found by van Wyk (2014),

an assumed mean yield stress of 687.5MPa is justified.

3.2.4 Signature Curves and Buckling Modes

The signature curves obtained by the CUFSM computer program for the calculation of Nd

and Nµ are shown in Figure 3.5 for the plain lipped C-section and in Figure 3.6 for the stiffened lipped C-section. The two signature curves of each section are similar in shape,

however the signature curve used to calculate Nµ is shifted lower than the signature curve

used to calculate Nd. The load factors corresponding to the signature curve minima for the Nd and Nµ signature curves are presented in Table 3.5 for the plain lipped C-section and

Table 3.6 for the stiffened lipped C-section.


49

Figure 3.5: Signature curves of plain lipped C-section

Figure 3.6: Signature curves of stiffened lipped C-section

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000 10000

Load

fact

or

Half wave-length (mm)

NC Signature Curve

NR Signature Curve

ll ld ll-g

Nd Signature Curve

Nµ Signature Curve

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10 100 1000 10000

Load

Fac

tor

Half wave-length (mm)

NC Signature Curve

NR Signature Curve

ll ld

Nd Signature Curve

Nµ Signature Curve

Í𝑁]^ 𝑁 _Î

Í𝑁]^ 𝑁 _Î

(a)

(b)

(c) (d)

(e)

(f) (g) (h)


50

Table 3.5: Load factors of Nd and Nµ signature curves for plain lipped-C section

Local buckling minima Distortional buckling minima

Signature

Curve

Load factor ¸tÏ,Ð¸Ì

Half-wavelength ll (mm)

Load factor ¸tÏ,Ñ¸Ì

Half-wavelength ld (mm)

Nd (c) 0.14955 68.33 (a) 0.33068 413.56

Nµ (d) 0.10767 68.33 (b) 0.23806 413.56

Table 3.6: Load factors of Nd and Nµ signature curves for stiffened lipped-C section

Local buckling minima Distortional buckling minima

Signature

Curve

Load factor ¸tÏ,Ð¸Ì

Half-wavelength ll (mm)

Load factor ¸tÏ,Ñ¸Ì

Half-wavelength ld (mm)

Nd (e) 0.096476 36.64 (g) 0.054979 453.57

Nµ (f) 0.069463 36.64 (h) 0.039585 453.57

The values in Table 3.5 and Table 3.6 correspond to those in Figure 3.5 and Figure 3.6

respectively. The load factors ¸tÏ,Ð¸_

and ¸tÏ,Ñ¸_

in Table 3.5 and Table 3.6 are ratios of the local

and distortional critical buckling loads respectively to the yield load. As seen in Figure

3.5 and Figure 3.6, these occur at the signature curve minima.

The DSM design equations suggest that, beyond a given buckling half-wavelength, a local-global buckling interaction occurs. This is to say that the calculated values of the local

buckling capacity Ncl and the elastic buckling Nce using the DSM are equal. Therefore, the reliability index at the local-global buckling interaction of the plain lipped C-section was

also investigated. This was done by obtaining the buckling half-wavelength in the global

buckling zone of the signature curve that shared an equal load factor at the local minimum on the signature curve, as shown in Figure 3.5. The symbol ll-g represents the local-global

buckling half-wavelength as shown on Figure 3.5. The buckling half-wavelength of the

local-global interaction was found to be 2024.61mm for the plain lipped C-section. No local-global buckling interaction was considered for the stiffened lipped C-section.


51

As observed for the plain lipped C-section, the local buckling load factor ¸tÏ,Ð¸Ì

is less than

the distortional buckling load factor ¸tÏ,Ñ¸Ì

for both signature curves of the plain lipped C-

section. This is shown in Table 3.5. This implies that local buckling will occur at a lower load than distortional buckling for the plain lipped C-section. Additionally, observed for

the stiffened lipped C-section, the distortional buckling load factor ¸tÏ,Ñ¸Ì

is less than the

local buckling load factor ¸tÏ,Ð¸Ì

for both signature curves of the stiffened lipped C-section.

This is shown in Table 3.6. This suggests that distortional buckling will occur at a lower load than local buckling for the stiffened lipped C-Section.

Global buckling is induced in both sections at higher buckling half-wavelengths. This is

to say that, irrespective of the section profile, the global buckling mode dominates the

buckling capacity of the members at high member lengths.

3.2.5 Direct Strength Method Design of Members

Following the design procedures of the DSM presented in Section 2.1.2.2.3, the DSM calculations of the local buckling capacity Ncl, the distortional buckling capacity Ncd and

elastic buckling capacity Nce were calculated for different member lengths. This was done

for the lipped and stiffened C-section members respectively, as shown in Figure 3.7 and Figure 3.8. For each member, the design, and the mean compressive loads to calculate Nd

and Nµ respectively are shown.

Additional to the figures, the buckling half-wavelengths for the curve minima in are also plotted in Figure 3.7 and Figure 3.8 respectively. The local and distortional buckling

minima plots on Figure 3.5 and Figure 3.6 are with reference to Figure 3.5 and Figure 3.6

respectively.


52

Figure 3.7: Direct strength method buckling loads for the plain lipped C-section

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 100 1000 10000

Com

pres

sive

Loa

d (N

)

Half-wavelength (mm)Nd Local Buckling NR Local Buckling

Nd Global Buckling NR Global Buckling

Nd Distortional Buckling NR Distortional Buckling

Local Buckling Minimum Distortional Buckling Minimum

Design Ncl

Design Nce

Design Ncd

Local Buckling Minimum

Mean Ncl

Mean Nce

Mean Ncd

Distortional Buckling Minimum


53

Figure 3.8: Direct strength method buckling loads for the stiffened lipped C-section

For the plain lipped C-section, it is observed that the local buckling minima of the signature curves in Figure 3.5 occurs at a buckling half-wavelength of 68.33mm.

However, when calculating the local buckling load of Nµ for the plain lipped C-section, the

design equations of the DSM show that distortional buckling dominates at 68.33mm and not local buckling. This is shown in Figure 3.7, since the distortional buckling DSM plot

is lower than the local buckling DSM plot at a length of 68.33mm. To avoid associating

the local buckling minimum on the signature curve with the distortional buckling load

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

10 100 1000 10000

Com

pres

sive

Loa

d (N

)

Half-wavelength (mm)

Nd Local Buckling NR Local BucklingNd Global Buckling NR Global BucklingNd Distortional Buckling NR Distortional BucklingLocal Buckling Minimum Distortional Buckling Minimum

Design Ncl

Design Nce

Design Ncd Local Buckling Minimum

Mean Ncl

Mean Nce

Mean Ncd Distortional Buckling Minimum


54

calculated with the DSM design equations, the local buckling load calculated from the DSM design equations was chosen for the local buckling analysis.

3.2.6 Reliability Analyses

As the main objective of this study was to determine the reliability index of the DSM, the

reliability index was determined for each of the buckling modes presented in the DSM design method. To perform a reliability analyses for each buckling mode, six reliability

analyses were performed. Four reliability analyses were performed on the plain lipped C-

section and two reliability analyses were performed on the stiffened lipped C-section. The motivation for these tests are subsequently explained.

3.2.6.1 Global Buckling Reliability Analysis The reliability analysis of the global buckling mode was performed in three tests. Test one was performed on the plain lipped C-section member with a length of 4000mm. This is

where the global buckling mode governs. Test two was performed on the plain lipped C-section member with a length of 2024.61mm. This is where the local-global buckling

interaction occurs, as the load factor at this point on the signature curve matches that at

the local buckling minimum. Test three was performed on the stiffened lipped C-section member with a length of 4000mm.

Comparing the results of test one and test two shows the significance of the length of the

member on the reliability levels, given it is subject to global buckling. Comparing the

results of test one and test three will show the significance of the shape of the section on the reliability levels, given that it is subject to global buckling. The global buckling model

factor presented in Table 3.13 was used in each of these three tests.

3.2.6.2 Local Buckling Reliability Analysis The reliability analysis of the local buckling mode was performed in a single test. Test

four was performed on the plain lipped C-section member with a length of 68.33mm. This

is the length of the member where the local buckling load factor ¸tÏ,Ð¸Ì

occurs on the

signature curve. The local-global buckling interaction model factor presented in Table

3.13 was used for the local buckling mode.

3.2.6.3 Local-Global Buckling Interaction Reliability Analysis The reliability analysis of the local-global buckling interaction was performed in a single

test. Test five again considers the plain lipped C-section member at a length of 2024.61mm like test 2. However, the local-global buckling interaction model factor presented in Table


55

3.13 was used for the reliability analysis for the buckling mode. This test assessed the local-global buckling interaction of the member by using the local-global buckling

interaction model factor for when the member is subject to global buckling.

3.2.6.4 Distortional Buckling Reliability Analysis The reliability analysis of the distortional buckling mode was performed in a single test.

Test 6 was performed on the stiffened lipped C-section member with a length of

453.57mm. This is where the distortional buckling load factor ¸tÏ,Ñ¸Ì

occurs on the signature

curve in Figure 3.6. The distortional buckling model factor presented in Table 3.13 for the

reliability analysis for distortional buckling.

3.2.6.5 Buckling Analysis Summary Table 3.7 summarises the six tests conducted to obtain the reliability analysis. It is to be

noted that for each of the tests, the reliability index as well as the associated sensitivity factors of the variables was obtained for each considered load combination. An explanation

of the model factors is presented in Section 3.4.2.

Table 3.7: Test summary of reliability analysis

Test Number Buckling Mode Section Type Resistance

Model Factor

Buckling half-

wavelength (mm)

1 Global Plain 𝜕R(g) 4000.00

2 Global Plain 𝜕R(g) 2024.61

3 Global Stiffened 𝜕R(g) 4000.00

4 Local Plain 𝜕R(lg) 68.33

5 Local-Global Plain 𝜕R(lg) 2024.61

6 Distortional Stiffened 𝜕R(d) 453.57

It is to be noted that test five is conducted to assess the reliability levels for the local-global buckling interaction. The test uses a local-global buckling interaction resistance

model factor for a member that is subject to global buckling. The reliability results of test

one and test five are compared. This is to assess whether the local-global buckling interaction or the global buckling mode dominates uncertainty for a member that is

subject to global buckling.


56

3.3 Semi-Probabilistic Formulation of the Limit-State Discussed in Section 3.1.1, the semi-probabilistic formulation of the limit-state was used

to obtain the characteristic values of the considered loads, for assumed load ratios, that would result in a column at its limit-state according to SANS 10162-2 (2011). This was

done by obtaining the design compressive critical buckling capacity Nd and equating it to

the total design load effect. The characteristic values of the considered loads would then be used in the full probabilistic formulation of the limit-state, as a basis to describe the

loads probabilistically.

3.3.1 Obtaining Characteristic Values of the Loads

Once the design capacity of the member was calculated using the equations of the DSM, the considered loads were determined. Subsequently, the characteristic values of the

considered loads were then calculated. Considering Equation 2.20, the limit-state is

reached when the design structural resistance is equal to the design load effect. This is detailed in Equation 3.4, assuming load combinations of permanent, imposed and wind

loading. Each design load may be expressed in terms of its characteristic loads and partial

factors, as shown in Equation 3.5. The introduction of the combination factors is discussed once the dominating load case is identified.

𝑁k = 𝐺k + 𝑄k +𝑊k (3.4)

= 𝛾¡ ∙ 𝐺Ò + 𝛾È ∙ 𝑄Ò + 𝛾Ó ∙ 𝑊Ò (3.5)

The symbols 𝛾G,𝛾Qand𝛾W are the partial factors of the permanent, imposed and the wind loads respectively. The partial factors are used to convert the characteristic values to the

design values of each of the load conditions. The values of the partial factors depend on

the limit-state and load combination scheme under consideration. Equation 3.5, however, does not consider action combination factors that are dependent on the dominating load

combination.

For the purposes of this study, four load combinations were considered for the analysis,

in accordance with the previsions of SANS 10160-1 (2011). Load combination 1 considers the STR limit-state with the imposed load as the leading load variable. Load combination

2 considers the STR limit-state with the wind load as the leading load variable. Load

combination 3 considers the STR-P limit-state with the imposed load as the leading load


57

variable. Load combination 4 considers the STR-P limit-state with the wind load as the

leading load variable. These are summarised in Table 3.8.

Table 3.8: Summary of considered load combinations

Load Combination Denotation Equation

1 STR:Q 𝑁k = 𝛾¡𝐺Ò + 𝛾È𝑄Ò + 𝛾�𝛹�𝑊Ò (3.6)

2 STR:W 𝑁k = 𝛾¡𝐺Ò + 𝛾È𝛹È𝑄Ò + 𝛾�𝑊Ò (3.7)

3 STR-P:Q 𝑁k = 𝛾¡𝐺Ò + 𝛾È𝑄Ò (3.8)

4 STR-P:W 𝑁k = 𝛾¡𝐺Ò + 𝛾Ó𝑊Ò (3.9)

Where 𝛹W and 𝛹Q are the action combination factors for the accompanying variable action

of wind load in STR:Q and imposed load in STR:W respectively. The values of the partial factors and the combination factors are dependent on the load combination and are given

in Table 3.9.

All partial factors and combination factors are obtained from SANS 10160-1 (2011) for

unfavourable load cases, except for the wind load partial factor for an STR limit state. SANS 10160-1 (2011) suggests a wind load partial factor value of 1.3 be used in design.

However, Botha (2016) recommends an update of the current wind load partial factor to

a value of 1.6. Therefore, the recommended updated value is used in this study.

Table 3.9: Values of partial and combination factors

Limit State

Variable STR STR-P

𝛾G 1.2 1.35

𝛾Q 1.6 1.0

𝛾W 1.6 1.0

𝛹Q 0.3 0

𝛹W 0 0


58

In SANS 10160-1 (2011), the values of the combination factors not only depend on the load

case but also the specific use of the structure. For the purposes of this study, it was

assumed that the structure type was of Category C: Public areas where people may congregate. The equations presented in Table 3.8 can be summarised as a conditional

general expression. The conditional general expression is expressed in Equation 3.10.

𝑁k = 𝛾¡𝐺Ò + 𝛾È𝛹È𝑄Ò + 𝛾�𝛹�𝑊Ò (3.10)

On the condition that

𝛹È =0.3 𝑓𝑜𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛20 𝑓𝑜𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛41 𝑓𝑜𝑟𝑜𝑡ℎ𝑒𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠

(3.11)

𝛹Ó =0 𝑓𝑜𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛10 𝑓𝑜𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛31 𝑓𝑜𝑟𝑜𝑡ℎ𝑒𝑟𝑙𝑜𝑎𝑑𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠

(3.12)

Since it was unclear as to which load combination dominated the load effect, two load

ratios were subsequently defined. The load ratios express the characteristic variable loads as a fraction of the total characteristic load. These are shown in Equations 3.13 and 3.14.

𝜒È =𝑄Ò

𝐺Ò + 𝑄Ò + 𝑊Ò(3.13)

𝜒Ó =𝑊Ò

𝐺Ò + 𝑄Ò + 𝑊Ò(3.14)

The symbol 𝜒Èis the imposed load ratio and 𝜒Ó is the wind load ratio. The values of the

characteristic load ratios were parametrically varied from 0 to 1, in increments of 0.1. The

reliability index was determined for each load combination of characteristic load ratios. Rearranging Equations 3.13 and 3.14 gives Equation 3.15 and 3.16 respectively.

𝜒È ∙ 𝐺Ò + 𝜒È − 1 ∙ 𝑄Ò + 𝜒È ∙ 𝑊Ò = 0 (3.15)

𝜒Ó ∙ 𝐺Ò + 𝜒Ó ∙ 𝑄Ò + 𝜒Ó − 1 ∙ 𝑊Ò = 0 (3.16)

Equations 3.10, 3.15 and 3.16 can be solved simultaneously in matrix form, as shown in Equation 3.17 to Equation 3.19, to obtain the characteristic loads {Nk}. The matrix form

expresses the characteristic loads as a vector and the associated factors as a 3 x 3 matrix.

This is expressed in symbolic form.


59

𝛾¡ 𝛾È𝛹È 𝛾�𝛹�𝜒È 𝜒È − 1 𝜒È𝜒Ó 𝜒Ó 𝜒Ó − 1

∙𝐺Ò𝑄Ò𝑊Ò

=𝑁k00

(3.17)

𝐴 ∙ 𝑁Ò = 𝑁k (3.18)

𝑁Ò = 𝑖𝑛𝑣 𝐴 ∙ 𝑁k (3.19)

In Equations 3.18 and 3.19, [A] is the load factor matrix, {Nk} is the characteristic load

vector and {Nd} is the design load vector.

3.3.2 Load Combination Choice

Each of the considered load combinations presented in Table 3.8 had a dominating effect

for a certain combination of 𝜒È and 𝜒Ó. The dominating load combination was determined

for each considered incremental value of 𝜒È and 𝜒Ó.

Considering a total characteristic unit load, the respective design loads for each load combination and set of load ratios were calculated. The calculation process to identify the

governing load combination for each value of 𝜒È and 𝜒Ó is subsequently summarised.


60

1. Consider a characteristic unit load (i.e.: Gk + Qk + Wk = 1)

2. It follows from Equations 3.13 and 3.14 that:

a. Qk = 𝜒Èb. Wk = 𝜒Óc. Gk = 1 −𝜒È −𝜒Ó

3. Initial conditions:

a. Set 𝜒È = 0

b. Set 𝜒Ó = 0

4. For 0 ≤ 𝜒È ≤ 1:

a. For 0 ≤ 𝜒Ó ≤ 1:

i. If 𝜒È + 𝜒Ó ≤ 1:

1. Compute corresponding total design load for each load

case2. Dominating load case: max[Ed(LC1), Ed(LC2), Ed(LC3), Ed(LC4)]

3. Increment 𝜒Ó by 0.1

4. Repeat loop

ii. Else if 𝜒È + 𝜒Ó > 1, end loop

b. Let 𝜒Ó = 0

c. Increment 𝜒È by 0.1

d. If 𝜒È + 𝜒Ó ≤ 1:

i. Repeat loop

e. Else if 𝜒È + 𝜒Ó > 1:

i. End loop, all dominating load combinations are determined

5. The dominating load combinations are found over the range of considered load

ratios and are shown in Table 3.10.


61

Table 3.10: Dominating load combinations for different values of χQ and χW

𝜒È

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

𝜒Ó

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 STR:Q dominates

0.8 STR:W dominates

0.9 STR-P:Q dominates

1.0 STR-P:W dominates

It is to be noted, with reference to Table 3.10, the hatched diagonal from 𝜒È = 0.0 and

𝜒� = 1.0 to 𝜒È = 1.0 and 𝜒È = 0.0 constitutes a theoretical load case where there is no dead

load. This is practically impossible, but it is still an important assessment at the

theoretical outer range of the code application.


62

3.4 Full Probabilistic Formulation of the Limit-State The full probabilistic formulation of the limit state expresses the variables of the load

effect and of the structural resistance as random variables. Additionally, the associated model factors were also considered as random variables. The reliability analysis was

conducted using a FORM analysis.

3.4.1 Variables of the Load Effect

Considered as separate continuous random variables, each of the load types were represented by probability density functions. Holický (2009) provides probabilistic

descriptions of the load types as shown in Table 3.11, expressed in terms of characteristic

values of each of the load types.

Table 3.11: Statistical moment parameters of the considered loads (Holický, 2009)

Load

Variable

Symbol

X

Distribution

function

Mean

Value

µX

Standard

Deviation

σX

Coefficient of

Variation (%)

V

Permanent G Normal Gk 0.07Gk 7

Imposed Q Gumbel 0.6Qk 0.21Qk 35

Wind W Gumbel 0.65Wk 0.32Wk 49

Holický (2009) suggests that the standard deviation of the permanent load may be

between 0.03Gk and 0.10Gk. The average of the range was used for this study.

When compared to the other loads, the permanent load has a significantly lower coefficient of variation. Additionally, the mean value of the permanent load is equal to the

characteristic value of the permanent load. This shows that the permanent load is more

accurately predicted than the other considered variable loads. Therefore, no model factor for the permanent load was considered in the full probabilistic formulation of the limit-

state.

The imposed load considered in this study corresponds to a 50-year reference period.

Unlike the permanent load, the imposed load has a relatively larger coefficient of variance and has a non-symmetrical Gumbel distribution. This implies that it may significantly


63

affect structural reliability, especially in cases with a high proportion of imposed load. A model factor is included for the imposed load. From Holický (2009), the model factor for

the imposed load used in this study has a normal probability distribution. The mean of

the model factor is 1.0 and the standard deviation may be between 0.05 to 0.10. In this study, the average of this range was used.

From the research of Botha (2016), the statistical moment parameters of the wind load

used in this study are based on the updated Eurocode wind load prediction model. The

statistical moment parameters of the wind load obtained from Botha (2016) accounts for the 50-year extremes of wind pressure, the pressure coefficient, the roughness factor and

the model coefficient. Having the highest coefficient of variance of the three considered

loads, the wind load may have the most significant effect on reliability. Table 3.12 shows the incorporated variables of the wind load model and their associated statistical moment

parameters (Botha, 2016). The relatively high coefficient of variance of the design wind load model is partly due to the model factor being incorporated in the moment parameters

of the wind load.

Table 3.12: Updated Eurocode full probabilistic wind load model of the FORM analysis (Botha, 2016)

Variable Distribution

Relative

Mean µ𝑿

𝑿𝒌

Standard

Deviation s𝑿

𝑿𝒌

Coefficient of

Variation (%)

V

50-year extremes of wind pressure

Gumbel 0.92 0.31 34

Pressurecoefficient Normal 1.00 0.16 16

Roughness factor Normal 0.84 0.10 12

ModelCoefficient Normal 0.80 0.16 20

Design wind load Gumbel 0.65 0.32 49


64

3.4.2 Variables of the Structural Resistance

In this study, the overall structural resistance R in the full probabilistic formulation of

the limit-state was composed of two variables; namely the mean critical buckling capacity

Nµ and the structural resistance model factor 𝜕R. This is shown in Equation 3.20.

𝑅 = 𝜕� ∙ 𝑁Ç (3.20)

In the calculation of the mean critical buckling capacity of the member, the mean values for member thickness material yield stress were used. This is to say that the material

thickness and the yield stress were taken as deterministic values. Therefore, the mean

critical buckling load was considered as a deterministic value. The element thickness and the material yield stress were not considered as random variables because the results of

Bauer (2016). The study found that the model factor for the structural resistance had a

dominating sensitivity contribution for the reliability index. Therefore, the structural resistance model factor was the only random variable in the calculation of the overall

structural resistance.

Bauer (2016) recommended that different structural resistance model factors be used for

different buckling modes. In a study by Ganesan and Moen (2010), the recorded data of 675 CFS column specimens were collected and analysed. The statistical moment

parameters of the model factors for the variety of considered sections were determined in accordance to the DSM. Subsequently, the results for the appropriate sections were

grouped in terms of the buckling modes considered in the DSM.

The shape of each of the distribution functions were not explicitly mentioned in Ganesan

and Moen (2010) and were therefore considered to be normally distributed, based on Holický (2009).

From the research of Ganesan and Moen (2010), the statistical moment parameters of the

model factors for each buckling mode are presented in Table 3.13.


65

Table 3.13: Statistical moment parameters of the structural resistance model factors for different buckling modes (Ganesan & Moen, 2010)

Model Factor Type Symbol

𝜕R

Distribution

function

Mean

Value

µX

Standard

Deviation

σX

Coefficient of

Variation (%)

V

Local-global

buckling

interaction

𝜕R(lg) Normal 1.03 0.15 14

Distortional

buckling 𝜕R(d) Normal 1.07 0.10 9

Global buckling or yielding

𝜕R(g) Normal 1.06 0.22 20

3.4.3 Reliability Analysis

The full probabilistic formulation of the limit-state equation for the reliability analysis

conducted in this study is expressed in Equation 3.21. VaP, a computer program that implements the FORM algorithm was used in this study. VaP is distributed by

Petschacher Software and Development (Petschacher, 1997) and is a well-established

program. The program allows for the definition of an implicit limit-state as a function of probabilistic variables. The input parameters of the probabilistic variables are the

distribution type and the statistical moment parameters.

𝐺 = ∂� ∙ 𝑁Ç − (𝐺 + ∂È ∙ 𝑄 + 𝑊) (3.21)

Assessments were conducted on the reliability levels for each of the dominating load

combinations. For all the reliability results, a check was done to assess whether they were greater than the target reliability level of βt = 3 presented in SANS 10160-1 (2011).

Additional to the reliability levels, the sensitivity factors were obtained of each of the

random variables for each dominating load combination. The sensitivity factors show the

influence that each of the random variables have on the reliability level.

Tests that yielded reliability results below that of the target reliability were assessed and scrutinized. Reasons for the unsatisfactory levels of reliability are subsequently discussed.


66

Additionally, the reliability levels achieved by the FORM analysis are validated by comparing those that are achieved by the MC analysis.

3.4.4 Monte-Carlo Simulation

As mentioned in Section 2.2.2.2, a MCS was performed to determine whether the limit-

state is concave or convex. The limit-state is concave if the reliability level achieved from the MCS is less than that achieved by the FORM analysis. Conversely, the limit-state is

convex if the reliability level achieved from the MCS is greater than that achieved by the

FORM analysis.

If the limit-state is concave, then the FORM analysis is an under-conservative. If the limit-state is convex, then the FORM analysis is conservative. A MCS was performed for

each dominating load combination to estimate the shape of the limit-state for each load

combination. Observing Table 3.10, four MC analyses were performed at the combinations

of 𝜒È and 𝜒Ó shown in Table 3.14.

Table 3.14: MC analyses of dominating load combinations

Dominating Load

Combination

Associated

Leading Variable

𝝌𝑸 𝝌𝑾

STR

STR

Q 1.0 0.0

W 0.0 1.0

STR-P

STR-P

Q 0.1 0.0

W 0.0 0.1

3.4.5 Chapter Summary

This chapter covered the necessary processes that were conducted in this study to assess

the inherent reliability of the SANS 10162-2 (2011) formulation of the DSM for

compression members. Two representative cold-formed steel members were considered in this study. A plain lipped C-section and a stiffened lipped C-Section were considered. The

local and global buckling modes were the dominating buckling modes for the plain lipped

C-section. The distortional and global buckling modes were the dominating buckling modes for the stiffened lipped C-section. The geometric and material properties of the

considered members adhered to the prequalification limitations of the DSM.


67

Signature curves for the mean and design compressive critical buckling loads were generated for each of the members. From the signature curves, the local, distortional, and

global buckling load factors, as well as the corresponding buckling half-wavelengths were

identified. The load factors and corresponding buckling half-wavelengths were used in the DSM design equations for each of the dominating critical buckling mode. The design and

mean compressive critical buckling capacity of the members was determined. The design

compressive critical buckling capacity was used in the semi-probabilistic formulation of the limit-state. The mean compressive buckling capacity was used in the full probabilistic

formulation of the limit-state.

The design and mean compressive critical buckling capacity of the members were

designed in accordance to the SANS 10162-2 (2011) formulation of the DSM. However, to achieve a true and unconservative representation of the member, the capacity reduction

factor was not considered for the calculation of the mean compressive critical buckling load. Additionally, the mean yield stress was used in the calculations of the mean

compressive critical buckling load.

Considering the semi-probabilistic formulation of the limit-state, the design compressive

critical buckling capacity of the member was equated to the codified design load effect. The load effect consisted of permanent, imposed and wind loads. The characteristic values

of the considered loads were obtained for a range of load ratios and load combinations in

accordance with SANS 10160-1 (2011).

Of the full probabilistic formulation of the limit-state, each variable was considered as a random variable. The structural resistance consisted of the model factor and the mean

compressive critical buckling load. The load effect consisted of permanent, imposed and

wind loads. Only the model factor for the imposed load was considered.

A FORM analysis was conducted. The results of the form analysis included the reliability levels achieved for each buckling mode, as well as the sensitivity factors for each

considered random variable. A MC analysis was conducted for each of the considered load

combinations to check the validity of the results of the FORM analysis. If the resulting reliability levels of each conducted test were below the target reliability presented in

SANS 10160-1 (2011) of βt = 3, there may be cause for concern.


68

4 CHAPTER 4: Results & Discussion

4.1 Results of Reliability Analysis This section shows the results of the full probabilistic formulation of the limit-state. As discussed in Section 3.2.6.5, six tests were conducted. For each of the tests, the reliability

level was assessed for all possible combinations of χÈ and χÓ. Additionally, the sensitivity

factors of each random variable in the full probabilistic formulation of the limit-state are

shown. The sensitivity factors give an indication of which of the random variables in the full probabilistic formulation of the limit-state have the largest influence on the level of

uncertainty for a given load combination.

For given load ratios of χÈ and χÓ where the reliability level is low, an analysis was made

to determine which of the random variables in the full probabilistic formulation of the

limit state equation had the greatest effect on uncertainty. This was done by observing

the sensitivity factors of each of the random variables for the given load ratios of χÈ and

χÓ.

As discussed in previous chapters, the sensitivity factors are either classified as having a minor, significant or a dominating effect, depending on its value. The reliability levels are

characterised in increments of 0.5.

4.1.1 Global Buckling

As discussed in Section 3.2.6.1, three tests were conducted to assess the reliability of the global buckling capacity of a member using the DSM. Test one was conducted on the plain

lipped C-section at a length of 4000mm. Test two was also conducted on the plain lipped

C-section but at a length of 2024.61mm. Test two was tested at the local-global interaction of the member, but the global buckling model factor was used. Test three was conducted

on the stiffened lipped C-section at a length of 4000mm. The global buckling model factor

was used for tests one to three.


69

4.1.1.1 Test One The following results are of the global buckling mode of the plain lipped C-section with a member length of 4000mm. Table 4.1 shows the reliability levels for the first test.

Accompanying these results are the sensitivity factors for each of the random variables

considered in the full probabilistic formulation of the limit-state equation. Table 4.2 to Table 4.6 show the sensitivity factors for the structural resistance model factor, the

imposed load model factor, the permanent load, the imposed load, and the wind load

respectively.

Table 4.1: Reliability levels for test one

𝜒È

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 1.92 1.97 2.02 2.21 2.38 2.53 2.64 2.73 2.80 2.84 2.87

0.1 1.96 1.78 1.86 2.08 2.27 2.42 2.55 2.64 2.71 2.75

0.2 1.98 1.95 1.91 1.89 2.10 2.27 2.41 2.51 2.58

0.3 2.14 2.12 2.08 2.03 1.95 2.06 2.22 2.34

0.4 2.26 2.24 2.20 2.16 2.09 2.00 1.96 β= 2.5 – 3.0

0.5 2.34 2.31 2.28 2.23 2.17 2.09 β=2.0 – 2.5

0.6 2.38 2.36 2.32 2.28 2.22 β=1.5 – 2.0

0.7 2.40 2.38 2.34 2.29

0.8 2.41 2.38 2.34

0.9 2.41 2.38

1.0 2.40


70

Table 4.2: α𝜕R for test one

𝜒È

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.981 0.982 0.978 0.971 0.957 0.934 0.900 0.857 0.806 0.755 0.707

0.1 0.979 0.977 0.973 0.964 0.947 0.919 0.880 0.830 0.775 0.723

0.2 0.968 0.965 0.957 0.940 0.921 0.890 0.847 0.796 0.741

0.3 0.944 0.938 0.926 0.905 0.872 0.839 0.800 0.753

0.4 0.903 0.891 0.874 0.851 0.819 0.777 0.730

0.5 0.845 0.827 0.806 0.780 0.751 0.715 Minor Effect

0.6 0.779 0.758 0.735 0.709 0.682 Significant Effect

0.7 0.715 0.694 0.671 0.647 Dominating Effect

0.8 0.660 0.639 0.617

0.9 0.613 0.593

1.0 0.574

Table 4.3: α𝜕Q for test one

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.000 0.014 0.029 0.044 0.060 0.078 0.095 0.113 0.128 0.142 0.152

0.1 0.000 0.015 0.032 0.049 0.066 0.085 0.103 0.121 0.136 0.149

0.2 0.000 0.015 0.032 0.052 0.071 0.091 0.110 0.127 0.142

0.3 0.000 0.014 0.030 0.050 0.072 0.093 0.112 0.130

0.4 0.000 0.013 0.027 0.045 0.065 0.088 0.109

0.5 0.000 0.011 0.024 0.039 0.056 0.076 Minor Effect



0.8 0.000 0.008 0.017

0.9 0.000 0.007

1.0 0.000


71

Table 4.4: αG for test one

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.196 0.182 0.165 0.139 0.114 0.090 0.068 0.047 0.029 0.013 0.000

0.1 0.181 0.179 0.159 0.130 0.103 0.078 0.054 0.033 0.015 0.000

0.2 0.164 0.151 0.137 0.118 0.089 0.062 0.038 0.017 0.000

0.3 0.135 0.122 0.106 0.089 0.069 0.044 0.020 0.000

0.4 0.108 0.094 0.078 0.060 0.041 0.021 0.000

0.5 0.082 0.067 0.052 0.036 0.018 0.000 Minor Effect



0.8 0.023 0.012 0.000

0.9 0.011 0.000

1.0 0.000

Table 4.5: αQ for test one

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.000 0.057 0.122 0.188 0.259 0.337 0.419 0.501 0.577 0.640 0.690

0.1 0.000 0.063 0.134 0.206 0.284 0.368 0.454 0.539 0.613 0.672

0.2 0.000 0.061 0.135 0.223 0.306 0.394 0.484 0.568 0.640

0.3 0.000 0.057 0.127 0.211 0.310 0.404 0.494 0.579

0.4 0.000 0.053 0.115 0.190 0.278 0.378 0.477

0.5 0.000 0.047 0.102 0.166 0.240 0.326 Minor Effect



0.8 0.000 0.033 0.069

0.9 0.000 0.030

1.0 0.000


72

Table 4.6: αW for test one

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.088 0.098 0.099 0.096 0.091 0.086 0.080 0.072 0.065 0.059

0.2 0.192 0.204 0.217 0.225 0.215 0.201 0.184 0.166 0.147

0.3 0.301 0.320 0.338 0.354 0.364 0.349 0.320 0.286

0.4 0.416 0.441 0.465 0.484 0.496 0.496 0.476

0.5 0.529 0.556 0.581 0.600 0.613 0.613 Minor Effect



0.8 0.751 0.768 0.783

0.9 0.790 0.804

1.0 0.819

Observing Table 4.1, the reliability level for all possible combinations of χÈ and χÓ are

below the minimum allowable level of reliability of βt=3, presented in SANS 10160-1

(2011) for the RC2 reliability class. The lowest recorded level of reliability is β=1.78,

where χÈ = 0.1 and χÓ = 0.1. This is where the STR-P:Q and STR-P:W load combinations

dominate. The highest recorded reliability level is β=2.87 where χÈ = 1.0 and χÓ = 0.0.

This is where the STR:Q load combination dominates.

For the load ratio of χÈ and χÓ where the reliability level is lowest, it is evident that the

sensitivity factor for the structural resistance model factor in Table 4.2 has a dominating

effect on the level of uncertainty. All the other random variables have a minor effect for this load combination.

4.1.1.2 Test Two The following results are of the global buckling mode of the plain lipped C-section with a

member length of 2024.61mm. The reliability levels for the second test are shown in Table

4.7. The sensitivity factors for each of the random variables in the full probabilistic formulation of the limit-state equation are shown in Table 4.8 to Table 4.12 for the


73

structural resistance model factor, the imposed load model factor, the permanent load, the imposed load, and the wind load respectively.

Table 4.7: Reliability levels for test two

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 1.92 1.97 2.02 2.21 2.38 2.53 2.64 2.73 2.80 2.84 2.87

0.1 1.96 1.78 1.86 2.08 2.26 2.42 2.54 2.64 2.71 2.75

0.2 1.98 1.95 1.91 1.89 2.10 2.27 2.41 2.51 2.58

0.3 2.14 2.12 2.08 2.03 1.95 2.06 2.22 2.34

0.4 2.26 2.24 2.20 2.15 2.09 2.00 1.96 β= 2.5 – 3.0

0.5 2.34 2.31 2.28 2.23 2.17 2.09 β=2.0 – 2.5

0.6 2.38 2.36 2.32 2.28 2.22 β=1.5 – 2.0

0.7 2.38 2.38 2.32 2.29

0.8 2.41 2.38 2.36

0.9 2.41 2.37

1.0 2.40


74

Table 4.8: α𝜕R for test two

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

χÓ

0.0 0.981 0.982 0.978 0.971 0.957 0.934 0.900 0.856 0.806 0.755 0.707

0.1 0.979 0.977 0.973 0.964 0.947 0.919 0.879 0.829 0.775 0.723

0.2 0.968 0.965 0.957 0.940 0.921 0.890 0.847 0.795 0.741

0.3 0.944 0.938 0.926 0.905 0.872 0.839 0.800 0.752

0.4 0.903 0.891 0.874 0.850 0.819 0.777 0.730

0.5 0.845 0.827 0.805 0.780 0.751 0.715 Minor Effect



0.8 0.659 0.639 0.615

0.9 0.613 0.593

1.0 0.574

Table 4.9: α𝜕Q for test two

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.014 0.029 0.044 0.060 0.078 0.095 0.113 0.129 0.142 0.152

0.1 0.000 0.015 0.032 0.049 0.066 0.085 0.103 0.121 0.136 0.149

0.2 0.000 0.015 0.032 0.052 0.071 0.091 0.110 0.128 0.142

0.3 0.000 0.014 0.030 0.050 0.072 0.093 0.113 0.130

0.4 0.000 0.013 0.027 0.045 0.065 0.088 0.109

χÓ 0.5 0.000 0.011 0.024 0.039 0.056 0.076 Minor Effect



0.8 0.000 0.008 0.016

0.9 0.000 0.007

1.0 0.000


75

Table 4.10: αG for test two

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.196 0.182 0.165 0.139 0.114 0.090 0.068 0.047 0.029 0.013 0.000

0.1 0.181 0.179 0.159 0.130 0.103 0.078 0.054 0.033 0.015 0.000

0.2 0.164 0.151 0.137 0.118 0.089 0.062 0.038 0.017 0.000

0.3 0.135 0.122 0.106 0.089 0.069 0.044 0.020 0.000

0.4 0.108 0.093 0.078 0.060 0.041 0.021 0.000

χÓ 0.5 0.082 0.067 0.052 0.036 0.018 0.000 Minor Effect



0.8 0.023 0.012 0.000

0.9 0.011 0.000

1.0 0.000

Table 4.11: αQ for test two

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.057 0.122 0.188 0.259 0.337 0.420 0.502 0.578 0.640 0.690

0.1 0.000 0.063 0.134 0.206 0.284 0.368 0.456 0.540 0.613 0.671

0.2 0.000 0.061 0.135 0.223 0.306 0.394 0.485 0.569 0.640

0.3 0.000 0.057 0.127 0.211 0.310 0.405 0.495 0.579

0.4 0.000 0.053 0.115 0.190 0.278 0.378 0.477

χÓ 0.5 0.000 0.047 0.102 0.166 0.240 0.325 Minor Effect



0.8 0.000 0.033 0.069

0.9 0.000 0.030

1.0 0.000


76

Table 4.12: αW for test two

χÈ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.088 0.098 0.099 0.096 0.091 0.086 0.079 0.072 0.065 0.061

0.2 0.192 0.206 0.217 0.225 0.215 0.201 0.184 0.165 0.147

0.3 0.301 0.320 0.339 0.355 0.365 0.349 0.320 0.285

0.4 0.416 0.442 0.465 0.485 0.497 0.496 0.476

χÓ 0.5 0.529 0.556 0.581 0.601 0.613 0.614 Minor Effect



0.8 0.751 0.768 0.785

0.9 0.790 0.805

1.0 0.819

Comparing the results in Table 4.1 and Table 4.7, the reliability levels of test one and test

two are similar. As in test one, the reliability levels for all possible combinations of χÈ and

χÓ are below the minimum allowable level of reliability of βt=3 for test two. Comparing

test one and test two, it is evident that the uncertainty levels of a member do not depend on the length of the member, given that the member is subject to global buckling.

4.1.1.3 Test Three The following results are of the global buckling mode of the stiffened lipped C-section with

a member length of 4000mm. Table 4.13 shows the reliability levels for the third test.

Table 4.14 to show the sensitivity factors for the structural resistance model factor, the imposed load model factor, the permanent load, imposed load, and the wind load

respectively.


77

Table 4.13: Reliability levels for test three

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 1.92 1.97 2.02 2.21 2.38 2.53 2.64 2.73 2.80 2.84 2.87

0.1 1.96 1.78 1.86 2.08 2.27 2.42 2.55 2.64 2.71 2.75

0.2 1.98 1.95 1.91 1.89 2.10 2.27 2.41 2.51 2.58

0.3 2.14 2.12 2.08 2.03 1.95 2.06 2.22 2.34

0.4 2.26 2.24 2.20 2.16 2.09 2.00 1.96 β = 2.5 – 3.0

χÓ 0.5 2.34 2.32 2.28 2.23 2.17 2.09 β = 2.0 - 2.5

0.6 2.38 2.36 2.32 2.28 2.22 β = 1.5 - 2.0

0.7 2.40 2.38 2.34 2.29

0.8 2.41 2.38 2.34

0.9 2.41 2.38

1.0 2.40

Table 4.14: α𝜕R for test three

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.981 0.982 0.978 0.971 0.957 0.934 0.900 0.857 0.806 0.755 0.708

0.1 0.979 0.977 0.973 0.964 0.947 0.919 0.879 0.830 0.775 0.723

0.2 0.968 0.965 0.957 0.940 0.921 0.890 0.847 0.796 0.741

0.3 0.944 0.938 0.926 0.905 0.872 0.839 0.800 0.753

0.4 0.903 0.891 0.874 0.851 0.819 0.777 0.730

χÓ 0.5 0.845 0.827 0.806 0.781 0.751 0.715 Minor Effect



0.8 0.660 0.639 0.617

0.9 0.613 0.593

1.0 0.574


78

Table 4.15: α𝜕Q for test three

χÓ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.014 0.029 0.044 0.060 0.078 0.095 0.113 0.128 0.142 0.152

0.1 0.000 0.015 0.032 0.049 0.066 0.085 0.103 0.121 0.136 0.149

0.2 0.000 0.015 0.032 0.052 0.071 0.091 0.110 0.127 0.142

0.3 0.000 0.014 0.030 0.050 0.072 0.093 0.112 0.130

0.4 0.000 0.013 0.027 0.045 0.065 0.088 0.109

χÓ 0.5 0.000 0.011 0.024 0.039 0.056 0.076 Minor Effect



0.8 0.000 0.008 0.017

0.9 0.000 0.007

1.0 0.000

Table 4.16: αG for test three

χÓ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.196 0.182 0.165 0.139 0.114 0.090 0.068 0.047 0.029 0.013 0.000

0.1 0.181 0.179 0.159 0.130 0.103 0.078 0.054 0.033 0.015 0.000

0.2 0.163 0.151 0.137 0.118 0.089 0.062 0.038 0.017 0.000

0.3 0.135 0.122 0.106 0.089 0.069 0.044 0.020 0.000

0.4 0.108 0.094 0.078 0.060 0.041 0.021 0.000

χÓ 0.5 0.082 0.067 0.052 0.036 0.018 0.000 Minor Effect



0.8 0.023 0.012 0.000

0.9 0.011 0.000

1.0 0.000


79

Table 4.17: αQ for test three

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.057 0.122 0.188 0.259 0.337 0.419 0.501 0.577 0.640 0.690

0.1 0.000 0.063 0.134 0.206 0.284 0.368 0.456 0.539 0.613 0.672

0.2 0.000 0.061 0.135 0.223 0.306 0.394 0.484 0.568 0.640

0.3 0.000 0.057 0.127 0.211 0.310 0.404 0.494 0.578

0.4 0.000 0.053 0.115 0.190 0.278 0.378 0.477

χÓ 0.5 0.000 0.047 0.102 0.166 0.240 0.326 Minor Effect



0.8 0.000 0.033 0.069

0.9 0.000 0.030

1.0 0.000

Table 4.18: αW for test three

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.088 0.098 0.099 0.096 0.091 0.086 0.079 0.072 0.065 0.059

0.2 0.192 0.204 0.217 0.225 0.215 0.201 0.184 0.166 0.147

0.3 0.300 0.320 0.338 0.354 0.364 0.349 0.320 0.286

0.4 0.416 0.441 0.464 0.484 0.496 0.496 0.477

χÓ 0.5 0.528 0.556 0.581 0.600 0.613 0.613 Minor Effect



0.8 0.751 0.768 0.784

0.9 0.790 0.804

1.0 0.819


80

The results presented in Table 4.13 are once again like those presented in Table 4.1. The reliability levels are below the minimum allowable level of reliability of βt=3 for all

possible combinations of χÈ and χÓ. Comparing the results of test three and test one, it is

evident that the sectional shape of the member does not influence the reliability levels,

given that the member is subject to global buckling and the cross-sectional area remains

constant.

4.1.1.4 Global Buckling Test Summary For all the tests conducted for the global buckling failure mode, the reliability level was consistent. This is observed when comparing the values in Table 4.1, Table 4.7 and Table

4.13. From this, two observations are made. Firstly, the length of the member does not

influence the levels of reliability or the sensitivity factors of the random variables, given the member is subject to global buckling. Secondly, the cross-sectional shape of the

member does not influence the levels of reliability or the sensitivity factors of the random

variables, given that the member is subject to global buckling.

Additionally, the sensitivity factors of the random variables in the full probabilistic formulation of the limit-state remained constant for different member lengths and section

shapes, provided global buckling governed the design capacity. For all three tests, the

structural resistance model factor had a dominating effect on the reliability level for the combination of load ratios that yielded the lowest reliability level. This is observed when

comparing the values in Table 4.2, Table 4.8 and Table 4.14.

For all the global buckling analyses, the imposed load sensitivity factor and the permanent load sensitivity factor have a minor effect on the reliability level for all possible

combinations of χÈ and χÓ . This is observed when comparing the values in Table 4.4,

Table 4.10 and Table 4.16. The sensitivity factors for the imposed load increase from a

minor effect to a significant effect as the proportion of imposed load increases. The sensitivity factors for the wind load increase from a minor effect to a dominating effect as

the proportion of wind load increases.

As mentioned in Section 3.4.3, MC analyses were conducted to determine whether the

limit-state for each load combination is concave or convex. The results are shown in Table 4.19.


81

Table 4.19: Reliability levels achieved through FORM and MCS for each load combination

Dominating Load

Combination

Associated

Leading Variable

𝝌𝑸 𝝌𝑾 FORM

β

MCS

β

STR

STR

Q 1.0 0.0 2.87 2.80

W 0.0 1.0 2.40 2.34

STR-P

STR-P

Q 0.1 0.0 1.97 1.96

W 0.0 0.1 1.96 1.95

Since none of the reliability levels achieved by the MCS analysis exceed those achieved

by the FORM analysis, the shape of the limit-state for each load combination is convex.

This implies that the FORM analysis is a conservative statistical analysis, specific to the tests conducted in this study.

4.1.2 Local Buckling and Local-Global Buckling Interaction

Test four was conducted to assess the reliability of the local buckling capacity of a member

and test five was conducted to assess the reliability of the local-global buckling interaction buckling capacity of the member. Test four was conducted on the plain lipped C-section

at a length of 68.33mm. Test five was also conducted on the plain lipped C-section but at

a length of 2024.61mm. The local-global buckling interaction model factor was used for both tests.

4.1.2.1 Test Four The following results are of the local buckling mode of the plain lipped C-section with a

member length of 68.33mm. Table 4.20 shows the reliability levels for test four. The

sensitivity factors for the structural resistance model factor, the imposed load model factor, the permanent load, the imposed load, and the wind load are shown in Table 4.21

to Table 4.25 respectively.


82

Table 4.20: Reliability levels for test four

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 3.35 3.42 3.46 3.67 3.80 3.86 3.86 3.82 3.78 3.73 3.69

0.1 3.40 3.18 3.26 3.50 3.63 3.69 3.69 3.66 3.62 3.59

0.2 3.39 3.34 3.27 3.20 3.37 3.45 3.47 3.46 3.44

0.3 3.48 3.43 3.35 3.23 3.08 3.11 3.19 3.21

0.4 3.46 3.40 3.32 3.22 3.09 2.92 2.80 β= 3.5 - 4.0 χÓ 0.5 3.38 3.32 3.24 3.15 3.04 2.90 β = 3.0 - 3.5

0.6 3.29 3.23 3.16 3.07 2.97 β = 2.5 - 3.0 0.7 3.21 3.15 3.08 3.00

0.8 3.14 3.08 3.01

0.9 3.07 3.01

1.0 3.01

Table 4.21: α𝜕R for test four

χÓ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.973 0.974 0.966 0.945 0.890 0.796 0.704 0.632 0.577 0.534 0.500

0.1 0.970 0.966 0.957 0.930 0.864 0.764 0.674 0.605 0.553 0.512

0.2 0.943 0.936 0.919 0.883 0.823 0.731 0.646 0.580 0.530

0.3 0.862 0.843 0.820 0.790 0.744 0.686 0.617 0.556

0.4 0.740 0.717 0.693 0.669 0.643 0.611 0.569

χÓ 0.5 0.640 0.619 0.599 0.578 0.558 0.537 Minor Effect



0.8 0.474 0.460 0.445

0.9 0.442 0.428

1.0 0.415


83

Table 4.22: α𝜕Q for test four

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.017 0.038 0.063 0.095 0.129 0.153 0.168 0.178 0.185 0.190

0.1 0.000 0.019 0.042 0.070 0.104 0.137 0.159 0.172 0.181 0.187

0.2 0.000 0.018 0.041 0.072 0.106 0.138 0.160 0.173 0.182

0.3 0.000 0.015 0.034 0.060 0.092 0.126 0.153 0.169

0.4 0.000 0.012 0.027 0.045 0.068 0.098 0.130

χÓ 0.5 0.000 0.010 0.022 0.036 0.053 0.074 Minor Effect



0.8 0.000 0.007 0.014

0.9 0.000 0.006

1.0 0.000

Table 4.23: αG for test four

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.233 0.215 0.195 0.162 0.127 0.092 0.063 0.041 0.025 0.011 0.000

0.1 0.214 0.212 0.186 0.150 0.113 0.077 0.049 0.029 0.013 0.000

0.2 0.190 0.175 0.157 0.132 0.095 0.061 0.035 0.015 0.000

0.3 0.148 0.131 0.113 0.092 0.070 0.043 0.018 0.000

0.4 0.105 0.090 0.074 0.057 0.039 0.020 0.000

χÓ 0.5 0.074 0.060 0.046 0.032 0.016 0.000 Minor Effect



0.8 0.020 0.010 0.000

0.9 0.009 0.000

1.0 0.000


84

Table 4.24: αQ for test four

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.070 0.163 0.277 0.427 0.584 0.691 0.755 0.796 0.825 0.845

0.1 0.000 0.078 0.179 0.306 0.468 0.619 0.716 0.774 0.811 0.837

0.2 0.000 0.074 0.175 0.315 0.476 0.627 0.724 0.782 0.818

0.3 0.000 0.064 0.146 0.258 0.409 0.569 0.693 0.767

0.4 0.000 0.052 0.115 0.194 0.298 0.436 0.582

χÓ 0.5 0.000 0.043 0.093 0.153 0.228 0.322 Minor Effect



0.8 0.000 0.029 0.060

0.9 0.000 0.026

1.0 0.000

Table 4.25: αW for test four

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.112 0.125 0.127 0.119 0.106 0.089 0.075 0.064 0.056 0.050

0.2 0.275 0.295 0.312 0.314 0.274 0.221 0.179 0.150 0.129

0.3 0.484 0.517 0.541 0.545 0.515 0.432 0.339 0.273

0.4 0.665 0.690 0.707 0.714 0.701 0.653 0.566

χÓ 0.5 0.765 0.782 0.794 0.800 0.796 0.776 Minor Effect



0.8 0.880 0.888 0.893

0.9 0.897 0.903

1.0 0.910


85

The results presented in Table 4.20 show that most of the possible combinations of χÈ and

χÓ adhere to the minimum level of reliability of 3. However, the reliability levels are below

the target reliability level of βt = 3, where the wind and imposed loads are similar and

dominate the load effect.

The lowest recorded reliability level is β=2.80, where χÈ = 0.6 and χÓ = 0.4. This is where

the STR:Q load combination dominates. The highest recorded level of reliability is

β=3.86, where χÈ = 0.5 or 0.6 and χÓ = 0.0. This is where the STR:Q load combination

dominates. For the load ratio of χÈ and χÓ where the reliability level is lowest, the

structural resistance model factor, the imposed load and the wind load have a significant

effect on the level of uncertainty.

4.1.2.2 Test Five The following results are of the local-global buckling interaction mode of the plain lipped

C-section with a member length of 2024.61mm. The reliability levels for test five are shown in Table 4.26. The corresponding sensitivity factors for the structural resistance

model factor, the imposed load model factor, the permanent load, the imposed load, and

the wind load are shown in Table 4.27 to Table 4.31 respectively.

Table 4.26: Reliability levels for test five

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 2.56 2.64 2.70 2.95 3.12 3.20 3.23 3.23 3.21 3.19 3.17

0.1 2.61 2.35 2.46 2.73 2.92 3.02 3.06 3.07 3.06 3.05

0.2 2.61 2.56 2.48 2.41 2.63 2.77 2.84 2.87 2.89

0.3 2.75 2.69 2.62 2.50 2.35 2.42 2.55 2.63

0.4 2.78 2.73 2.65 2.55 2.42 2.25 2.17 β = 3.0 - 3.5

χÓ 0.5 2.75 2.69 2.62 2.54 2.43 2.31 β = 2.5 - 3.0

0.6 2.70 2.65 2.58 2.51 2.42 β = 2.0 - 2.5

0.7 2.66 2.60 2.54 2.48

0.8 2.61 2.56 2.51

0.9 2.57 2.52

1.0 2.61


86

Table 4.27: α𝜕R for test five

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.959 0.961 0.951 0.923 0.857 0.759 0.668 0.598 0.543 0.501 0.467

0.1 0.956 0.950 0.938 0.905 0.832 0.732 0.643 0.575 0.522 0.481

0.2 0.921 0.913 0.892 0.852 0.790 0.702 0.619 0.553 0.502

0.3 0.833 0.815 0.791 0.759 0.711 0.654 0.590 0.532

0.4 0.756 0.736 0.713 0.683 0.646 0.596 0.544

χÓ 0.5 0.614 0.594 0.575 0.555 0.534 0.511 Minor Effect



0.8 0.447 0.433 0.419

0.9 0.415 0.402

1.0 0.392

Table 4.28: α𝜕Q for test five

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.020 0.045 0.074 0.108 0.138 0.159 0.173 0.181 0.187 0.192

0.1 0.000 0.022 0.049 0.080 0.114 0.144 0.163 0.176 0.184 0.189

0.2 0.000 0.021 0.048 0.080 0.114 0.143 0.163 0.175 0.184

0.3 0.000 0.018 0.040 0.068 0.100 0.130 0.154 0.170

0.4 0.000 0.016 0.034 0.057 0.083 0.111 0.136

χÓ 0.5 0.000 0.012 0.026 0.042 0.061 0.084 Minor Effect



0.8 0.000 0.008 0.017

0.9 0.000 0.007

1.0 0.000


87

Table 4.29: αG for test five

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.282 0.261 0.236 0.194 0.150 0.108 0.074 0.048 0.028 0.013 0.000

0.1 0.260 0.256 0.225 0.180 0.133 0.091 0.058 0.034 0.015 0.000

0.2 0.228 0.210 0.187 0.156 0.112 0.072 0.041 0.018 0.000

0.3 0.175 0.155 0.133 0.109 0.082 0.050 0.022 0.000

0.4 0.132 0.113 0.093 0.071 0.048 0.024 0.000

χÓ 0.5 0.087 0.071 0.054 0.037 0.019 0.000 Minor Effect



0.8 0.023 0.012 0.000

0.9 0.011 0.000

1.0 0.000

Table 4.30: αQ for test five

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.085 0.194 0.323 0.482 0.627 0.723 0.781 0.819 0.845 0.863

0.1 0.000 0.093 0.209 0.349 0.511 0.651 0.741 0.795 0.830 0.854

0.2 0.000 0.088 0.208 0.349 0.504 0.645 0.738 0.796 0.832

0.3 0.000 0.075 0.170 0.292 0.438 0.579 0.694 0.768

0.4 0.000 0.065 0.145 0.243 0.361 0.487 0.603

χÓ 0.5 0.000 0.050 0.109 0.179 0.262 0.363 Minor Effect



0.8 0.000 0.033 0.070

0.9 0.000 0.030

1.0 0.000


88

Table 4.31: αW for test five

χÓ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.134 0.148 0.150 0.141 0.125 0.104 0.087 0.075 0.065 0.058

0.2 0.317 0.338 0.352 0.349 0.310 0.255 0.208 0.174 0.149

0.3 0.525 0.553 0.571 0.568 0.535 0.467 0.382 0.314

0.4 0.641 0.664 0.679 0.682 0.666 0.628 0.568

χÓ 0.5 0.784 0.799 0.809 0.810 0.801 0.775 Minor Effect



0.8 0.894 0.901 0.905

0.9 0.910 0.915

1.0 0.920

The results presented in Table 4.26 show that most of the possible combinations of χÈ and

χÓ do not achieve the minimum level of reliability βt = 3. Scenarios where there is a higher

portion of imposed load meets the target level of reliability.

The lowest assessed level of reliability is 2.17, at χÈ = 0.6 and χÓ = 0.4, where the STR:Q

load combination dominates. The highest recorded level of reliability is β = 3.23, at

χÈ = 0.6 or 0.7 and χÓ = 0.0, where the STR:Q load combination dominates. For the load

ratio of χÈ and χÓ where the reliability level is lowest, the structural resistance model

factor, the imposed load, and the wind load each have significant effects on the level of

uncertainty. However, this is at the theoretical limit where there is no imposed load and the scenario is practically impossible.

4.1.2.3 Local Buckling and Local-Global Buckling Interaction Test Summary When comparing the reliability results of test four and test five, it is evident that the

lowest level of reliability is reached for both tests at the same load ratio of χÈ = 0.6 and

χÓ = 0.4. This is observed in Table 4.20 and Table 4.26. For both tests, the sensitivity

factors of the structural resistance model factor, the imposed load and the wind load have

significant effects on the level of reliability where the reliability level is at its lowest. This


89

is observed in Table 4.21 and Table 4.27. It is to be noted that the STR:Q load combination yielded the lowest levels of reliability for both tests.

The reliability levels achieved from the local-global buckling mode are higher than those

of the global buckling mode and lower than the local buckling mode. This is expected

because, as mentioned in Section 2.1.2.2.1, an interaction of buckling modes reduces the section capacity.

For all the local buckling analyses, the sensitivity factors for the imposed load model

factor and the permanent load have a minor effect on uncertainty for all possible

combinations of χÈ and χÓ. This is observed in Table 4.22 and Table 4.23 respectively. The

sensitivity factors for the imposed load increases from a minor effect to a dominating effect as the proportion of imposed load increases. The sensitivity factors for the wind load

increase from a minor effect to a dominating effect as the proportion of wind load

increases. This is observed in Table 4.25.

4.1.3 Distortional Buckling

As discussed in Section 3.2.6.4, one test was conducted to assess the reliability of the

distortional buckling capacity of a member using the DSM. Test six was conducted on the

stiffened lipped C-section at a length of 453.57mm. The distortional buckling modal factor was used for this test

4.1.3.1 Test Six The following results are of the distortional buckling mode of the stiffened lipped C-section

with a member length of 453.57mm. The reliability levels for test six are shown in Table

4.32. The corresponding sensitivity factors for the structural resistance model factor, the imposed load model factor, the permanent load, the imposed load, and the wind load are

shown in Table 4.33 to Table 4.37 respectively.


90

Table 4.32: Reliability levels for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 4.88 4.99 4.95 4.88 4.67 4.47 4.30 4.16 4.05 3.95 3.87

0.1 4.92 4.55 4.58 4.55 4.38 4.21 4.07 3.95 3.86 3.77

0.2 4.55 4.43 4.25 4.01 3.99 3.89 3.79 3.71 3.63

0.3 4.21 4.09 3.95 3.76 3.51 3.46 3.45 3.41

0.4 3.92 3.81 3.69 3.54 3.37 3.15 2.99 β = 4.5 - 5.0

χÓ 0.5 3.69 3.59 3.49 3.36 3.22 3.06 β = 4.0 - 4.5

0.6 3.51 3.43 3.33 3.22 3.10 β = 3.5 - 4.0

0.7 3.38 3.29 3.21 3.11 β = 3.0 - 3.5

0.8 3.26 3.19 3.11 β = 2.5 - 3.0

0.9 3.17 3.10

1.0 3.09

Table 4.33: α𝜕R for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.934 0.935 0.864 0.653 0.529 0.455 0.406 0.369 0.341 0.318 0.300

0.1 0.919 0.905 0.835 0.635 0.514 0.442 0.392 0.357 0.329 0.307

0.2 0.700 0.679 0.659 0.615 0.505 0.431 0.381 0.345 0.318

0.3 0.529 0.514 0.499 0.486 0.470 0.423 0.373 0.336

0.4 0.439 0.427 0.415 0.403 0.392 0.380 0.356

χÓ 0.5 0.383 0.371 0.361 0.351 0.340 0.331 Minor Effect



0.8 0.290 0.282 0.274

0.9 0.272 0.264

1.0 0.256


91

Table 4.34: α𝜕Q for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.029 0.090 0.164 0.191 0.202 0.208 0.211 0.213 0.214 0.214

0.1 0.000 0.032 0.093 0.164 0.190 0.201 0.206 0.209 0.211 0.212

0.2 0.000 0.022 0.055 0.124 0.177 0.194 0.202 0.206 0.208

0.3 0.000 0.015 0.035 0.063 0.111 0.166 0.188 0.198

0.4 0.000 0.012 0.026 0.044 0.069 0.105 0.148

χÓ 0.5 0.000 0.010 0.021 0.035 0.052 0.075 Minor Effect



0.8 0.000 0.007 0.014

0.9 0.000 0.006

1.0 0.000

Table 4.35: αG for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.358 0.331 0.279 0.179 0.121 0.084 0.058 0.039 0.023 0.011 0.000

0.1 0.325 0.318 0.261 0.164 0.107 0.071 0.046 0.027 0.012 0.000

0.2 0.226 0.203 0.180 0.147 0.093 0.058 0.033 0.014 0.000

0.3 0.145 0.128 0.110 0.091 0.071 0.042 0.018 0.000

0.4 0.100 0.086 0.071 0.055 0.038 0.020 0.000

χÓ 0.5 0.071 0.058 0.045 0.031 0.016 0.000 Minor Effect



0.8 0.020 0.010 0.000

0.9 0.009 0.000

1.0 0.000


92

Table 4.36: αQ for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.126 0.410 0.717 0.818 0.863 0.888 0.904 0.916 0.924 0.930

0.1 0.000 0.135 0.421 0.723 0.823 0.868 0.892 0.908 0.919 0.926

0.2 0.000 0.091 0.240 0.562 0.785 0.853 0.885 0.904 0.917

0.3 0.000 0.063 0.148 0.275 0.502 0.750 0.843 0.882

0.4 0.000 0.050 0.111 0.191 0.303 0.471 0.670

χÓ 0.5 0.000 0.041 0.090 0.150 0.226 0.326 Minor Effect



0.8 0.000 0.028 0.059

0.9 0.000 0.025

1.0 0.000

Table 4.37: αW for test six

χÈ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.1 0.222 0.246 0.223 0.142 0.104 0.083 0.070 0.061 0.054 0.048

0.2 0.677 0.700 0.688 0.519 0.297 0.215 0.171 0.144 0.124

0.3 0.836 0.846 0.846 0.822 0.714 0.478 0.339 0.267

0.4 0.893 0.899 0.900 0.892 0.865 0.789 0.634

χÓ 0.5 0.921 0.926 0.927 0.923 0.911 0.883 Minor Effect



0.8 0.957 0.959 0.960

0.9 0.962 0.964

1.0 0.967


93

The results presented in Table 4.32 show that all but one of the possible combinations of

χÈ and χÓ meets the target level of reliability of βt = 3. Of all the tests conducted, test six

yielded the highest reliability levels for all possible combinations of χÈ and χÓ.

The lowest recorded level of reliability was β=2.99, where χÈ = 0.6 and χÓ = 0.4. This is

where the STR:Q load combination dominates. The highest recorded level of reliability

was β=4.99, where χÈ = 0.1 and χÓ = 0.0. This is where the STR-P:Q load combination

dominates. For the load ratio of χÈ and χÓ where the reliability level is lowest, the

structural resistance model factor, the imposed load and the wind load have a significant effect on the level of uncertainty.

4.1.3.2 Distortional Buckling Test Summary The distortional buckling analysis yielded the highest reliability levels of the six tests.

This is observed in Table 4.32. Only one combination of load ratios, where χÈ = 0.6 and

χÓ = 0.4, yielded a reliability level below the target reliability level. Since this is a

theoretical case where there is no permanent load, the case is practically impossible.

Therefore, the reliability levels of the distortional buckling capacity are generally acceptable.


94

4.2 Discussion of Reliability Results 4.2.1 Overview

The results of this study show the reliability levels of the local, distortional, and global

buckling modes. A total of six tests were conducted. Three tests were conducted to assess the reliability levels of the global buckling mode. Two tests were conducted to assess that

reliability levels of the local buckling mode; one of which assessed the reliability of the

local-global buckling mode interaction. One test was conducted to assess the reliability levels of the distortional buckling mode.

Of each test, the sensitivity factors of each of the associated random variables were also

determined. This was to assess the influence each of the random variables had on the

level of uncertainty.

4.2.2 Discussion of Reliability Levels of Buckling Modes

4.2.2.1 Distortional Buckling Mode Almost all the reliability levels for the distortional buckling mode were greater than the

target reliability of βt = 3. Only one load ratio yielded a reliability level below 3. At the

point of lowest reliability, the imposed and wind load ratios were χÈ = 0.6 and χÓ = 0.4

respectively, as seen in Table 4.32. However, this combination of the load ratios implies

that there is no permanent load. Since this scenario is impossible as permanent load is always present, there is no cause for concern of the reliability levels for the distortional

buckling capacity.

4.2.2.2 Local Buckling Mode Like the reliability levels of the distortional buckling mode, most of the reliability levels

of the local buckling mode were greater than the target reliability of βt = 3. The load ratios

that yielded unsatisfactory reliability levels were at low proportions of permanent load,

and where χÈ and χÓ are similar as seen in Table 4.20. The same sentiment of the

assessment of the reliability levels for distortional buckling mode applies to those of the

local buckling mode.

4.2.2.3 Local-Global Buckling Interaction The local-global buckling mode interaction yielded most reliability levels that were below

the target reliability level of βt = 3. As stated by Kwon et al. (2009), the interaction of

buckling modes significantly reduces the compressive capacity of the section. Although


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the reliability levels of the local-global buckling interaction are higher than those of the global buckling mode, they are lower than those of the local buckling mode.

4.2.2.4 Global Buckling Mode For the global buckling mode, the reliability levels generally prove to be significantly too

low. None of the load ratios yielded a reliability level greater than the target reliability

level of βt = 3. The lowest reliability level occurred where there was a significantly higher

proportion of permanent load with low proportions of imposed and wind loads. Of all the buckling modes, the global buckling mode yielded the lowest reliability levels.

4.2.3 Detailed Discussion

Unlike the global buckling mode, the results of the local and distortional buckling modes

show that the lowest levels of reliability occur with load cases of a low permanent load and similar imposed and wind load factors. The sensitivity factors that have the largest

effect on the reliability levels for these load cases are those of the structural resistance

model factor, the imposed load, and the wind load. However, the lowest reliability level occurs for the practically impossible load case where there is little to no permanent load.

The reliability results of the local-global interaction show that there are several load

combinations that yield unsatisfactory results. However, the results of the global buckling

mode yielded lower reliability levels. Therefore, the global buckling mode dominates uncertainty for members subjected to global buckling

It is cause for concern that, of all the considered buckling modes, the results of the global

buckling mode yielded unsatisfactory reliability levels for all the considered load

combinations. Observing the sensitivity factors, the global buckling model factor has a predominant effect on the level of reliability, especially for load combinations with high

permanent loads. Therefore, the predictive model for the global buckling capacity is poor.

These results suggest that the predictive capacity of the distortional and local buckling modes better predict their associated modes of failure than that of the global buckling

mode. Additionally, for the global buckling mode, the sensitivity factors of the considered

load effects are lower than those of the structural resistance model factor for load combinations that yielded low reliability levels. This is to say that the safety margin

inherent of SANS 10160-1 does not compensate enough to achieve an overall acceptable

level of reliability.


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The considered column was unbraced along the length of the member. However, this study did not consider the case where a slender column member is braced at points along its

length. For example, cladding may be fastened to the flange or web along the length of

the member with a bolt spacing. This supplies the member with braced or lateral stability.

Lateral stability along the length of a CFS column will affect the buckling modes and their respective load factors when compared to an unbraced CFS column. Depending on the bolt

spacing, global buckling is less likely to govern the failure mode, since the effective length

of the member about the one axis is decreased. In this case, distortional buckling may not be induced, since rotation of the top flange is restricted. Local buckling is therefore more

likely to govern. This may result in an increased reliability of the member.


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5 CHAPTER 5: Conclusion & Recommendations

5.1 Conclusion The reliability analysis conducted in this thesis investigated whether the inherent reliability of the SANS 10162-2 (2011) formulation of the DSM for compression members

adhered to the target level of reliability required by SANS 10160-1 (2011). The target level

of reliability is βt = 3 for a RC2 reliability classification. A full probabilistic formulation of

the limit-state was used to evaluate the reliability of the semi-probabilistic formulation of the limit-state as per SANS 10160-1 (2011) for the load effect and SANS 10162-2 (2011)

for structural resistance.

A FORM analysis was used to determine the reliability level of each buckling mode. A

plain lipped C-section was considered as a codified representative member adhering to the geometric and material property limitations of the DSM. The global and local buckling

capacities of the considered member were assessed. A lipped C-section with a web stiffener

was used to assess the distortional and global buckling capacities. The four load combinations of SANS 10160-1 (2011) for STR and STR-P, respectively with wind or

variable loads as leading actions were considered in the analysis. The total load was distributed into permanent, imposed and wind loads according to load ratios which was

parametrically varied to cover the range of application.

The distortional buckling mode yielded the highest reliability levels for all combinations

of the imposed and wind load ratios. All but one combination of the load ratios was above the target reliability level of βt = 3. The reliability levels for the distortional buckling mode

ranged from β=2.99 to β=4.99.

Most of the reliability levels for the local buckling mode were above the target reliability level of βt = 3. Load conditions with a low ratio of permanent load to the total load, and

similar levels of imposed and wind loads yielded the lowest reliability levels. The

reliability levels for the local buckling mode ranged from β = 2.80 to β = 3.86.


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Few of the reliability levels of the local-global buckling interaction were above that of the target reliability level of βt = 3. Load conditions with higher proportions of imposed load

yielded reliability levels above the target reliability of βt = 3. The reliability levels of the

local-global buckling interaction ranged from β= 2.17 to β = 3.23.

Of all buckling modes, it was found that the global buckling mode yielded the lowest

reliability levels. The reliability levels for all combinations of the imposed and wind load

ratios were below the target reliability level. The reliability levels for the global buckling mode ranged from β = 1.78 to β = 2.87. For a member subject to global buckling at a length

of 2024.61mm, the global buckling mode dominates uncertainty over the local-global

buckling interaction.

The length of the member and shape of the cross section influences the capacity of the

member. This is because ranges of member lengths are subjected to different buckling modes. However, for a given cross-sectional area, it was found that the shape of the cross

section does not influence the reliability level of the global buckling mode. Additionally,

the length of the member does not influence the reliability level given that the member length is within the global buckling range.

It was found that, for all possible combinations of the imposed and wind load ratios where

the reliability was below the minimum level of reliability, the structural resistance model factor governed uncertainty.

The study conducted by Bauer (2016) found that the safety margin supplied by the

resistance standard SANS 10162-2 (2011) is not enough to achieve acceptable levels of

reliability. In the conclusion of Bauer (2016), it was assumed that, by incorporating the loading standard, overall acceptable levels of reliability could be achieved. The main

conclusion of this study is that the safety margin provided by the loading standard of

SANS 10160-1 (2011) is not sufficient to make up for the lack of safety margin provided by SANS 10162-2 (2011) and to ensure an overall acceptable level of reliability. Thus, the

capacity reduction factor used in the DSM of SANS 10162-2 (2011) needs to be decreased

to achieve an overall acceptable level of reliability, particularly for cases where global buckling is the critical failure mode.


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5.2 Recommendations This study only considered the analysis of pure compression members. As part of a structural system, most columns are subject to compression and bending stresses

simultaneously. Therefore, the work presented in this study is limited. A further study

may need to be conducted, assessing the reliability levels of CFS beam-column members. Literature by Ganesan and Moen (2010) presents model factors for the buckling modes of

only pure compression members. Therefore, model factors for the buckling modes of

members subject to bending needs to be developed in order to analyse the reliability levels of the interaction of bending and compressive stresses in CFS members.

Additionally, incorporating the combination of bending and compressive stresses in CFS

member design using the DSM has not been entirely investigated. Currently, it is

recommended that the interaction of bending and compression forces for the DSM is to follow the basic methodology of the EWM through interaction equations. However, studies

are being conducted to investigate the true behaviour of members subject to the

interaction. A more thorough stability analysis of beam-columns may lead to a behaviour different to that assumed by the interaction equations. (Schafer, 2006b).

In this study, one capacity reduction factor as per SANS 10162-2 (2011) was used for all

buckling modes, as the considered cross sections adhered to the geometric and material

limitations of the DSM. Therefore, since it was found that the assessed reliability levels differed systematically for each buckling mode, it is recommended that the capacity

reduction factor for compression members φc proposed in SANS 10162-2 (2011) be re-

evaluated. Specifically, it is recommended that a capacity reduction factor be assigned for

each buckling mode.

A further reliability analysis was conducted for the global buckling mode, since it yielded

the lowest reliability levels of the buckling modes. The capacity reduction factor φc was

decreased from a value of 0.85 accordingly until the lowest level of reliability of βt = 3 was

achieved. It was found that, by using a capacity reduction of 0.58, the reliability analysis

yielded satisfactory reliability levels. The results show that the global buckling prediction model is poor.

Of all the buckling mode interactions, this study only considered the local-global buckling

interaction. Further research may need to be conducted to assess the influence that other


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buckling mode interactions have on the reliability levels. Additionally, the buckling interactions are not defined in the DSM.


101

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Documents

An Assessment of the Inherent Reliability of SANS 10162-2